沙普利-福克曼引理 (Chinese Wikipedia)

Analysis of information sources in references of the Wikipedia article "沙普利-福克曼引理" in Chinese language version.

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Guesnerie (1989，第138頁) Guesnerie, Roger. First-best allocation of resources with nonconvexities in production [生產非凸的資源之最優分配]. Cornet, Bernard; Tulkens, Henry (编). Contributions to Operations Research and Economics: The twentieth anniversary of CORE (Papers from the symposium held in Louvain-la-Neuve, January 1987) [運籌學與經濟學投稿：二十周年（[[新魯汶]]1987年舉辦討論會的論文集）]. Cambridge, Mass.: MIT Press. 1989: 99–143. ISBN 0-262-03149-3. MR 1104662 （英语）. 网址－维基内链冲突 (帮助)
(Ekeland 1999，第357–359頁): 1976年首部英文版中，埃科蘭德在附錄證明沙普利-福克曼引理，並在p. 373提到勒馬雷沙爾的實驗觀察。 Ekeland, Ivar. Appendix I: An a priori estimate in convex programming [附錄一：凸規劃的先驗估計]. Ekeland, Ivar; Temam, Roger (编). Convex analysis and variational problems [凸分析與變分問題]. Classics in Applied Mathematics 28 Corrected reprinting of the North-Holland. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). 1999: 357–373 [1976]. ISBN 0-89871-450-8. MR 1727362 （英语）.
Bertsekas (1996，第364–381頁)於p. 374引用Ekeland (1999)，並在p. 381引用Aubin & Ekeland (1976)：
Bertsekas, Dimitri P. 5.6 Large scale separable integer programming problems and the exponential method of multipliers [第5.6節：大規模可分整數規劃問題及乘子指數法]. Constrained optimization and Lagrange multiplier methods [受限優化和拉格朗日乘子法] 1982年Academic Press版的重印. Belmont, Mass.: Athena Scientific. 1996: xiii+395. ISBN 1-886529-04-3. MR 0690767 （英语）.

Bertsekas (1996，第364–381頁)將拉格朗日對偶法用到發電排程（英语：Scheduling (production processes)）上（即機組排程問題（英语：power system simulation）），此種問題有變量限制為整數，所以非凸：

Bertsekas, Dimitri P.; Lauer, Gregory S.; Sandell, Nils R., Jr.; Posbergh, Thomas A. Optimal short-term scheduling of large-scale power systems [大規模電力系統的最優短期排程] (PDF). IEEE Transactions on Automatic Control. January 1983, 28 (1): 1–11 [2 February 2011]. doi:10.1109/tac.1983.1103136. （原始内容存档 (PDF)于2021-09-09）（英语）. Proceedings of 1981 IEEE Conference on Decision and Control, San Diego, CA, December 1981, pp. 432–443.
Ekeland, Ivar. Appendix I: An a priori estimate in convex programming [附錄一：凸規劃的先驗估計]. Ekeland, Ivar; Temam, Roger (编). Convex analysis and variational problems [凸分析與變分問題]. Classics in Applied Mathematics 28 Corrected reprinting of the North-Holland. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). 1999: 357–373 [1976]. ISBN 0-89871-450-8. MR 1727362 （英语）.
Artstein & Vitale (1975，第881–882頁): Artstein, Zvi; Vitale, Richard A. A strong law of large numbers for random compact sets [隨機緊集的強大數定律]. The Annals of Probability. 1975, 3 (5): 879–882. JSTOR 2959130. MR 0385966. Zbl 0313.60012. doi:10.1214/aop/1176996275 （英语）.
Carter (2001，第93–94頁)，取n = 3。 Carter, Michael. Foundations of mathematical economics [數理經濟學基礎]. Cambridge, Mass.: MIT Press. 2001: xx+649. ISBN 0-262-53192-5. MR 1865841. (作者網站有習題題解). （原始内容存档于15 September 2006）（英语）.
Arrow & Hahn (1980，第375頁) Arrow, Kenneth J.; Hahn, Frank H. General competitive analysis [一般競爭分析]. Advanced Textbooks in Economics 12 San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6之重印版. Amsterdam: North-Holland. 1980 [1971]. ISBN 0-444-85497-5. MR 0439057 （英语）.
Rockafellar (1997，第10頁) Rockafellar, R. Tyrrell. Convex analysis [凸分析]. Princeton Landmarks in Mathematics. Princeton, N.J.: Princeton University Press. 1997: xviii+451. ISBN 0-691-01586-4. MR 1451876 （英语）. 1970年(MR 274683) Princeton Mathematical Series 28的重印版。
Arrow & Hahn (1980，第376頁)、Rockafellar (1997，第10–11頁)、Green & Heller (1981，第37頁) Arrow, Kenneth J.; Hahn, Frank H. General competitive analysis [一般競爭分析]. Advanced Textbooks in Economics 12 San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6之重印版. Amsterdam: North-Holland. 1980 [1971]. ISBN 0-444-85497-5. MR 0439057 （英语）. Rockafellar, R. Tyrrell. Convex analysis [凸分析]. Princeton Landmarks in Mathematics. Princeton, N.J.: Princeton University Press. 1997: xviii+451. ISBN 0-691-01586-4. MR 1451876 （英语）. 1970年(MR 274683) Princeton Mathematical Series 28的重印版。 Green, Jerry; Heller, Walter P. 1 Mathematical analysis and convexity with applications to economics [第1章：數學分析與凸性，及在經濟學之應用]. Arrow, Kenneth Joseph; Intriligator, Michael D (编). Handbook of mathematical economics, Volume I. Handbooks in Economics 1. Amsterdam: North-Holland Publishing Co. 1981: 15–52. ISBN 0-444-86126-2. MR 0634800. doi:10.1016/S1573-4382(81)01005-9 （英语）.
Arrow & Hahn (1980，第385頁)及Rockafellar (1997，第11–12頁) Arrow, Kenneth J.; Hahn, Frank H. General competitive analysis [一般競爭分析]. Advanced Textbooks in Economics 12 San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6之重印版. Amsterdam: North-Holland. 1980 [1971]. ISBN 0-444-85497-5. MR 0439057 （英语）. Rockafellar, R. Tyrrell. Convex analysis [凸分析]. Princeton Landmarks in Mathematics. Princeton, N.J.: Princeton University Press. 1997: xviii+451. ISBN 0-691-01586-4. MR 1451876 （英语）. 1970年(MR 274683) Princeton Mathematical Series 28的重印版。
Schneider (1993，第xi頁)及Rockafellar (1997，第16頁) Schneider, Rolf. Convex bodies: The Brunn–Minkowski theory [凸體：布倫-閔可夫斯基理論]. Encyclopedia of Mathematics and its Applications 44. Cambridge, UK: Cambridge University Press. 1993: xiv+490. ISBN 0-521-35220-7. MR 1216521 （英语）. Rockafellar, R. Tyrrell. Convex analysis [凸分析]. Princeton Landmarks in Mathematics. Princeton, N.J.: Princeton University Press. 1997: xviii+451. ISBN 0-691-01586-4. MR 1451876 （英语）. 1970年(MR 274683) Princeton Mathematical Series 28的重印版。
Rockafellar (1997，第17頁)及Starr (1997，第78頁) Rockafellar, R. Tyrrell. Convex analysis [凸分析]. Princeton Landmarks in Mathematics. Princeton, N.J.: Princeton University Press. 1997: xviii+451. ISBN 0-691-01586-4. MR 1451876 （英语）. 1970年(MR 274683) Princeton Mathematical Series 28的重印版。 Starr, Ross M. 8 Convex sets, separation theorems, and non-convex sets in R^N (new chapters 22 and 25–26 in (2011) second ed.) [第8章：R^N中的凸集、分離定理、非凸集（及2011年第二版新增的第22、25、26諸章）]. General equilibrium theory: An introduction [一般均衡理論：導論] First. Cambridge, UK: Cambridge University Press. 1997: xxiii+250. ISBN 0-521-56473-5. MR 1462618 （英语）.
Schneider (1993，第2–3頁) Schneider, Rolf. Convex bodies: The Brunn–Minkowski theory [凸體：布倫-閔可夫斯基理論]. Encyclopedia of Mathematics and its Applications 44. Cambridge, UK: Cambridge University Press. 1993: xiv+490. ISBN 0-521-35220-7. MR 1216521 （英语）.
Arrow & Hahn (1980，第387頁) Arrow, Kenneth J.; Hahn, Frank H. General competitive analysis [一般競爭分析]. Advanced Textbooks in Economics 12 San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6之重印版. Amsterdam: North-Holland. 1980 [1971]. ISBN 0-444-85497-5. MR 0439057 （英语）.
Schneider (1993，第131頁) Schneider, Rolf. Convex bodies: The Brunn–Minkowski theory [凸體：布倫-閔可夫斯基理論]. Encyclopedia of Mathematics and its Applications 44. Cambridge, UK: Cambridge University Press. 1993: xiv+490. ISBN 0-521-35220-7. MR 1216521 （英语）.
Schneider (1993，第140頁)歸功於Borwein & O'Brien (1978): Borwein, J. M.; O'Brien, R. C. Cancellation characterizes convexity [抵消之事，可以刻畫凸性]. Nanta Mathematica (Nanyang University). 1978, 11: 100–102. ISSN 0077-2739. MR 0510842 （英语）. Schneider, Rolf. Convex bodies: The Brunn–Minkowski theory [凸體：布倫-閔可夫斯基理論]. Encyclopedia of Mathematics and its Applications 44. Cambridge, UK: Cambridge University Press. 1993: xiv+490. ISBN 0-521-35220-7. MR 1216521 （英语）.
Schneider (1993，第129頁) Schneider, Rolf. Convex bodies: The Brunn–Minkowski theory [凸體：布倫-閔可夫斯基理論]. Encyclopedia of Mathematics and its Applications 44. Cambridge, UK: Cambridge University Press. 1993: xiv+490. ISBN 0-521-35220-7. MR 1216521 （英语）.
Schneider (1993，第129–130頁) Schneider, Rolf. Convex bodies: The Brunn–Minkowski theory [凸體：布倫-閔可夫斯基理論]. Encyclopedia of Mathematics and its Applications 44. Cambridge, UK: Cambridge University Press. 1993: xiv+490. ISBN 0-521-35220-7. MR 1216521 （英语）.
Arrow & Hahn (1980，第392–395頁) Arrow, Kenneth J.; Hahn, Frank H. General competitive analysis [一般競爭分析]. Advanced Textbooks in Economics 12 San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6之重印版. Amsterdam: North-Holland. 1980 [1971]. ISBN 0-444-85497-5. MR 0439057 （英语）.
Schneider (1993，第128頁) Schneider, Rolf. Convex bodies: The Brunn–Minkowski theory [凸體：布倫-閔可夫斯基理論]. Encyclopedia of Mathematics and its Applications 44. Cambridge, UK: Cambridge University Press. 1993: xiv+490. ISBN 0-521-35220-7. MR 1216521 （英语）.
Ekeland (1999，第357–359頁) Ekeland, Ivar. Appendix I: An a priori estimate in convex programming [附錄一：凸規劃的先驗估計]. Ekeland, Ivar; Temam, Roger (编). Convex analysis and variational problems [凸分析與變分問題]. Classics in Applied Mathematics 28 Corrected reprinting of the North-Holland. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). 1999: 357–373 [1976]. ISBN 0-89871-450-8. MR 1727362 （英语）.
Artstein (1980，第180頁) Artstein, Zvi. Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points [離散與連續砰砰及面空間，又或：找極值點]. SIAM Review. 1980, 22 (2): 172–185. JSTOR 2029960. MR 0564562. doi:10.1137/1022026 （英语）.
Starr, Ross M. Approximation of points of convex hull of a sum of sets by points of the sum: An elementary approach [以集合之和的點迫近和集凸包的點：初等進路]. Journal of Economic Theory. 1981, 25 (2): 314–317. MR 0640201. doi:10.1016/0022-0531(81)90010-7 （英语）.
Mas-Colell (1985，第58–61頁) and Arrow & Hahn (1980，第76–79頁) Arrow, Kenneth J.; Hahn, Frank H. General competitive analysis [一般競爭分析]. Advanced Textbooks in Economics 12 San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6之重印版. Amsterdam: North-Holland. 1980 [1971]. ISBN 0-444-85497-5. MR 0439057 （英语）.
Arrow & Hahn (1980，第79–81頁) Arrow, Kenneth J.; Hahn, Frank H. General competitive analysis [一般競爭分析]. Advanced Textbooks in Economics 12 San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6之重印版. Amsterdam: North-Holland. 1980 [1971]. ISBN 0-444-85497-5. MR 0439057 （英语）.
Diewert (1982，第552–553頁) Diewert, W. E. 12 Duality approaches to microeconomic theory [第12章：微觀經濟學的對偶進路]. Arrow, Kenneth Joseph; Intriligator, Michael D (编). Handbook of mathematical economics, Volume II [數理經濟學手冊・卷二]. Handbooks in Economics 1. Amsterdam: North-Holland Publishing Co. 1982: 535–599 [2021-08-31]. ISBN 978-0-444-86127-6. MR 0648778. doi:10.1016/S1573-4382(82)02007-4. （原始内容存档于2012-12-10）（英语）.
Wold (1943b，第231 and 239–240頁): Wold, Herman. A synthesis of pure demand analysis II [純需求分析綜論二]. Skandinavisk Aktuarietidskrift [斯堪的納維亞精算期刊]. 1943b, 26: 220–263. MR 0011939. doi:10.1080/03461238.1943.10404737.
Wold & Juréen (1953，第146頁): Wold, Herman; Juréen, Lars (in association with Wold). 8 Some further applications of preference fields (pp. 129–148) [八、偏好域的其他應用]. Demand analysis: A study in econometrics [需求分析：計量經濟學研究]. Wiley publications in statistics. New York: John Wiley and Sons, Inc. 1953: xvi+358. MR 0064385 （英语）.
Samuelson (1950，第359–360頁):
會注意到，競爭市場中，不能觀測到無差異曲線凸處（而不是凹）的任何點。此種點被永恆的黑暗遮蔽，除非我等令該消費者壟斷買方，且從非常凸的「預算曲線」上，選取所買的商品。（其沿此曲線，影響所買商品的價格。）在买方垄断的情況，仍可從均衡點觀測到的限制的斜算，推斷該人無差異曲線的斜率。［譯按：此處凸與凹的約定，與本條目相反。］
Samuelson, Paul A. The problem of integrability in utility theory [效用論的可積性問題]. Economica. New Series. November 1950, 17 (68): 355–385. JSTOR 2549499. MR 0043436. doi:10.2307/2549499 （英语）.
「永恆的黑暗」描述彌爾頓所著《失樂園》中的地獄，其卷二第592至594行將地獄的凹陷與塞波尼斯大沼澤（英语：Serbonian Bog）相比：
A gulf profound as that Serbonian Bog
Betwixt Damiata and Mount Casius old,
Where Armies whole have sunk.
彌爾頓對凹陷的描寫，是Arrow & Hahn (1980，第169頁)第7章"Markets with non-convex preferences and production"［非凸偏好與生產的市場］的題辭（英语：epigraph (literature)）。該章講解Starr (1969)的成果。 Arrow, Kenneth J.; Hahn, Frank H. General competitive analysis [一般競爭分析]. Advanced Textbooks in Economics 12 San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6之重印版. Amsterdam: North-Holland. 1980 [1971]. ISBN 0-444-85497-5. MR 0439057 （英语）. Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Arrow & Hahn (1980，第182頁) Arrow, Kenneth J.; Hahn, Frank H. General competitive analysis [一般競爭分析]. Advanced Textbooks in Economics 12 San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6之重印版. Amsterdam: North-Holland. 1980 [1971]. ISBN 0-444-85497-5. MR 0439057 （英语）.
Aumann (1966，第1–2頁): Aumann, Robert J. Existence of competitive equilibrium in markets with a continuum of traders [市場有連續統多個交易人，則存在競爭均衡]. Econometrica. January 1966, 34 (1): 1–17. JSTOR 1909854. MR 0191623. doi:10.2307/1909854 （英语）. Aumann (1966)用到 Aumann （1964, 1965）的結果：
Aumann, Robert J. Markets with a continuum of traders [連續統多個交易人的市場]. Econometrica. January–April 1964, 32 (1–2): 39–50. JSTOR 1913732. MR 0172689. doi:10.2307/1913732 （英语）.
Aumann, Robert J. Integrals of set-valued functions [集合值函數的積分]. Journal of Mathematical Analysis and Applications. August 1965, 12 (1): 1–12. MR 0185073. doi:10.1016/0022-247X(65)90049-1  （英语）.
據Diewert (1982，第552頁)所言，Wold (1943b，第243頁)及Wold & Juréen (1953，第146頁)已於較早前討論過取非凸偏好的凸包。
Diewert, W. E. 12 Duality approaches to microeconomic theory [第12章：微觀經濟學的對偶進路]. Arrow, Kenneth Joseph; Intriligator, Michael D (编). Handbook of mathematical economics, Volume II [數理經濟學手冊・卷二]. Handbooks in Economics 1. Amsterdam: North-Holland Publishing Co. 1982: 535–599 [2021-08-31]. ISBN 978-0-444-86127-6. MR 0648778. doi:10.1016/S1573-4382(82)02007-4. （原始内容存档于2012-12-10）（英语）.
Arrow & Hahn (1980，第169–182頁)、Starr (1969，第27–33頁) Arrow, Kenneth J.; Hahn, Frank H. General competitive analysis [一般競爭分析]. Advanced Textbooks in Economics 12 San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6之重印版. Amsterdam: North-Holland. 1980 [1971]. ISBN 0-444-85497-5. MR 0439057 （英语）. Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Green & Heller (1981，第44頁) Green, Jerry; Heller, Walter P. 1 Mathematical analysis and convexity with applications to economics [第1章：數學分析與凸性，及在經濟學之應用]. Arrow, Kenneth Joseph; Intriligator, Michael D (编). Handbook of mathematical economics, Volume I. Handbooks in Economics 1. Amsterdam: North-Holland Publishing Co. 1981: 15–52. ISBN 0-444-86126-2. MR 0634800. doi:10.1016/S1573-4382(81)01005-9 （英语）.
Guesnerie (1989，第99頁) Guesnerie, Roger. First-best allocation of resources with nonconvexities in production [生產非凸的資源之最優分配]. Cornet, Bernard; Tulkens, Henry (编). Contributions to Operations Research and Economics: The twentieth anniversary of CORE (Papers from the symposium held in Louvain-la-Neuve, January 1987) [運籌學與經濟學投稿：二十周年（[[新魯汶]]1987年舉辦討論會的論文集）]. Cambridge, Mass.: MIT Press. 1989: 99–143. ISBN 0-262-03149-3. MR 1104662 （英语）. 网址－维基内链冲突 (帮助)
Varian (1992，第393–394頁): Varian, Hal R. 21.2 Convexity and size [21.2節：凸性與大小]. Microeconomic Analysis [微觀經濟分析] 3rd. W. W. Norton & Company. 1992. ISBN 978-0-393-95735-8. MR 1036734 （英语）.
Mas-Colell, Whinston & Green (1995，第627–630頁): Mas-Colell, Andreu; Whinston, Michael D.; Green, Jerry R. 17.1 Large economies and nonconvexities [第17.1節：大經濟體與非凸性]. Microeconomic theory [微觀經濟理論]. Oxford University Press. 1995. ISBN 978-0-19-507340-9 （英语）.
Arrow & Hahn (1980，第169–182頁)

Mas-Colell (1985，第52–55, 145–146, 152–153, and 274–275頁): Mas-Colell, Andreu. 1.L Averages of sets [第1.L節：集合的平均]. The Theory of general economic equilibrium: A differentiable approach [一般經濟均衡理論：可微分的進路]. Econometric Society monographs 9. Cambridge University Press. 1985. ISBN 0-521-26514-2. MR 1113262 （英语）.

Hildenbrand (1974，第37, 115–116, 122, and 168頁): Hildenbrand, Werner. Core and equilibria of a large economy [大經濟體的核和均衡]. Princeton studies in mathematical economics 5. Princeton, N.J.: Princeton University Press. 1974: viii+251. ISBN 978-0-691-04189-6. MR 0389160 （英语）.
Arrow, Kenneth J.; Hahn, Frank H. General competitive analysis [一般競爭分析]. Advanced Textbooks in Economics 12 San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6之重印版. Amsterdam: North-Holland. 1980 [1971]. ISBN 0-444-85497-5. MR 0439057 （英语）.
Starr (1997，第169頁)，及
Ellickson (1994，第xviii, 306–310, 312, 328–329, 347, and 352頁): Ellickson, Bryan. Competitive equilibrium: Theory and applications [競爭均衡：理論及應用]. Cambridge University Press. 1994. ISBN 978-0-521-31988-1. doi:10.2277/0521319889 （英语）.
Starr, Ross M. 8 Convex sets, separation theorems, and non-convex sets in R^N (new chapters 22 and 25–26 in (2011) second ed.) [第8章：R^N中的凸集、分離定理、非凸集（及2011年第二版新增的第22、25、26諸章）]. General equilibrium theory: An introduction [一般均衡理論：導論] First. Cambridge, UK: Cambridge University Press. 1997: xxiii+250. ISBN 0-521-56473-5. MR 1462618 （英语）.
Ichiishi (1983，第24–25頁): Ichiishi, Tatsuro. Game theory for economic analysis [經濟分析的賽局理論]. Economic theory, econometrics, and mathematical economics. New York: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]. 1983: x+164. ISBN 0-12-370180-5. MR 0700688 （英语）.
Cassels (1981，第127 and 33–34頁): Cassels, J. W. S. Appendix A Convex sets [附錄A：凸集]. Economics for mathematicians [數學家的經濟學]. London Mathematical Society lecture note series 62. Cambridge, UK: Cambridge University Press. 1981: xi+145. ISBN 0-521-28614-X. MR 0657578 （英语）.
Aubin (2007，第458–476頁): Aubin, Jean-Pierre. 14.2 Duality in the case of non-convex integral criterion and constraints (especially 14.2.3 The Shapley–Folkman theorem, pages 463–465) [第14.2節：非凸積分準則和限制下的對偶性（尤其第14.2.3小節：沙普利-福克曼定理，pp. 463–465]. Mathematical methods of game and economic theory [賽局與經濟理論的數學方法] 連新序言，重印1982年North-Holland修訂英文版. Mineola, N.Y.: Dover Publications, Inc. 2007: xxxii+616. ISBN 978-0-486-46265-3. MR 2449499 （英语）.
Carter (2001，第93–94, 143, 318–319, 375–377, and 416頁) Carter, Michael. Foundations of mathematical economics [數理經濟學基礎]. Cambridge, Mass.: MIT Press. 2001: xx+649. ISBN 0-262-53192-5. MR 1865841. (作者網站有習題題解). （原始内容存档于15 September 2006）（英语）.
Trockel (1984，第30頁): Trockel, Walter. Market demand: An analysis of large economies with nonconvex preferences [市場需求：分析非凸偏好的大經濟體]. Lecture Notes in Economics and Mathematical Systems 223. Berlin: Springer-Verlag. 1984: viii+205. ISBN 3-540-12881-6. MR 0737006 （英语）.
Rockafellar (1997，第23頁) Rockafellar, R. Tyrrell. Convex analysis [凸分析]. Princeton Landmarks in Mathematics. Princeton, N.J.: Princeton University Press. 1997: xviii+451. ISBN 0-691-01586-4. MR 1451876 （英语）. 1970年(MR 274683) Princeton Mathematical Series 28的重印版。
某集合內的序列的極限，必在該集合的閉包中，即最小而包含原集合的閉集。兩個閉集的閔氏和不必閉，故若以 $\mathrm {Cl}$ 表示閉包，則雖然有包含關係
$\mathrm {Cl} (P)+\mathrm {Cl} (Q)\subseteq \mathrm {Cl} (P+Q),$
但可能為嚴格包含（左右不必相等）。據Rockafellar (1997，第49及75頁)，即使兩個被加項各已是閉凸集，仍可能是嚴格包含。若要使閔氏和變成閉，則要取其閉包，即要添加所有收斂序列的極限。 Rockafellar, R. Tyrrell. Convex analysis [凸分析]. Princeton Landmarks in Mathematics. Princeton, N.J.: Princeton University Press. 1997: xviii+451. ISBN 0-691-01586-4. MR 1451876 （英语）. 1970年(MR 274683) Princeton Mathematical Series 28的重印版。
Lemaréchal (1973，第38頁): Lemaréchal, Claude. Utilisation de la dualité dans les problémes non convexes [在非凸問題使用對偶] (报告). Domaine de Voluceau, Rocquencourt, Le Chesnay, France: IRIA（現INRIA）, Laboratoire de recherche en informatique et automatique: 41. April 1973 （法语）. |issue=被忽略 (帮助) 勒馬雷沙爾的實驗，日後有下列論文討論：
Aardal (1995，第2–3頁): Aardal, Karen. Optima interview Claude Lemaréchal [Optima訪問克勞德·勒馬雷沙爾] (PDF). Optima: Mathematical Programming Society Newsletter. March 1995, 45: 2–4 [2 February 2011]. （原始内容存档 (PDF)于2021-09-09）（英语）.

Hiriart-Urruty & Lemaréchal (1993，第143–145, 151, 153, and 156頁): Hiriart-Urruty, Jean-Baptiste; Lemaréchal, Claude. XII Abstract duality for practitioners [第十二章：實踐用的抽象對偶性]. Convex analysis and minimization algorithms, Volume II: Advanced theory and bundle methods [凸分析和最小化算法，第二卷：進階理論及束法]. Grundlehren der Mathematischen Wissenschaften [數理科學的基本原理] 306. Berlin: Springer-Verlag. 1993: 136–193 （及pp. 334–335所列的文獻附註）. ISBN 3-540-56852-2. MR 1295240 （英语）.
Ekeland, Ivar. Une estimation a priori en programmation non convexe [非凸規劃中的先驗估計]. Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences [（法國）科學院週刊]. Séries A et B. 1974, 279: 149–151. ISSN 0151-0509. MR 0395844 （法语）.
Aubin & Ekeland (1976，第226, 233, 235, 238, and 241頁): Aubin, J. P.; Ekeland, I. Estimates of the duality gap in nonconvex optimization [估計非凸優化的對偶間隙]. Mathematics of Operations Research. 1976, 1 (3): 225–245. JSTOR 3689565. MR 0449695. doi:10.1287/moor.1.3.225 （英语）.
Aubin & Ekeland (1976)及Ekeland (1999，第362–364頁)也考慮非凸最小值問題的閉凸包，即對原問題的蓋圖（英语：epigraph (mathematics)）取閉凸包所得的新問題。Di Guglielmo推廣到研究非凸多目標優化問題的擬凸閉包，即對目標函數的下水平集取凸閉包所得的問題：
Di Guglielmo (1977，第287–288頁): Di Guglielmo, F. Nonconvex duality in multiobjective optimization [多目標優化的非凸對偶]. Mathematics of Operations Research. 1977, 2 (3): 285–291. JSTOR 3689518. MR 0484418. doi:10.1287/moor.2.3.285 （英语）.

Ekeland, Ivar. Appendix I: An a priori estimate in convex programming [附錄一：凸規劃的先驗估計]. Ekeland, Ivar; Temam, Roger (编). Convex analysis and variational problems [凸分析與變分問題]. Classics in Applied Mathematics 28 Corrected reprinting of the North-Holland. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). 1999: 357–373 [1976]. ISBN 0-89871-450-8. MR 1727362 （英语）.
Schneider & Weil (2008，第45頁): Schneider, Rolf; Weil, Wolfgang. Stochastic and integral geometry [隨機與積分幾何]. Probability and its applications. Springer. 2008. ISBN 978-3-540-78858-4. MR 2455326. doi:10.1007/978-3-540-78859-1 （英语）.
Cassels (1975，第433–434頁): Cassels, J. W. S. Measures of the non-convexity of sets and the Shapley–Folkman–Starr theorem [集合的非凸度與沙普利-福克曼-斯塔定理]. Mathematical Proceedings of the Cambridge Philosophical Society. 1975, 78 (3): 433–436. MR 0385711. doi:10.1017/S0305004100051884 （英语）.
Molchanov (2005，第195–198, 218, 232, 237–238 and 407頁): Molchanov, Ilya. 3 Minkowski addition [第3章：閔可夫斯基加法]. Theory of random sets [論隨機集]. Probability and its applications. London: Springer-Verlag London Ltd. 2005: 194–240. ISBN 978-1-84996-949-9. MR 2132405. doi:10.1007/1-84628-150-4 （英语）.
Puri & Ralescu (1985，第154–155頁): Puri, Madan L.; Ralescu, Dan A. Limit theorems for random compact sets in Banach space [巴拿赫空間隨機緊集的極限定理]. Mathematical Proceedings of the Cambridge Philosophical Society. 1985, 97 (1): 151–158. Bibcode:1985MPCPS..97..151P. MR 0764504. doi:10.1017/S0305004100062691 （英语）.
Weil (1982，第203, and 205–206頁): Weil, Wolfgang. An application of the central limit theorem for Banach-space–valued random variables to the theory of random sets [取值於巴拿赫空間隨機變量的中央極限定理，應用於隨機集理論]. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete [概率論與相關領域期刊]. 1982, 60 (2): 203–208. MR 0663901. doi:10.1007/BF00531823 （英语）.
Cerf (1999，第243–244頁): Cerf, Raphaël. Large deviations for sums of i.i.d. random compact sets [獨立同分佈隨機緊集之和的大離差]. Proceedings of the American Mathematical Society. 1999, 127 (8): 2431–2436. MR 1487361. doi:10.1090/S0002-9939-99-04788-7  （英语）. 瑟夫（Cerf）用到 Puri & Ralescu (1985，第154–155頁)應用沙普利-福克曼引理的結果。
Ruzsa (1997，第345頁): Ruzsa, Imre Z. The Brunn–Minkowski inequality and nonconvex sets [布倫-閔可夫斯基不等式與非凸集]. Geometriae Dedicata. 1997, 67 (3): 337–348. MR 1475877. doi:10.1023/A:1004958110076 （英语）.
Tardella (1990，第478–479頁): Tardella, Fabio. A new proof of the Lyapunov convexity theorem [李亞普諾夫凸性定理的新證]. SIAM Journal on Control and Optimization. 1990, 28 (2): 478–481. MR 1040471. doi:10.1137/0328026 （英语）.
Artstein (1980，第172–183頁) Artstein (1980)在致敬2008年諾貝爾經濟學獎得主羅伯特·奧曼的論文集重新出版：Artstein, Zvi. 22 Discrete and continuous bang–bang and facial spaces or: Look for the extreme points [第22篇：離散與連續砰砰及面空間，又或：找極值點]. Hart, Sergiu; Neyman, Abraham (编). Game and economic theory: Selected contributions in honor of Robert J. Aumann [賽局與經濟理論：致敬羅伯特·J. 奧曼的文選]. Ann Arbor, Mich.: University of Michigan Press. 1995: 449–462. ISBN 0-472-10673-2. （原始内容存档于24 May 2011）（英语）. Artstein, Zvi. Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points [離散與連續砰砰及面空間，又或：找極值點]. SIAM Review. 1980, 22 (2): 172–185. JSTOR 2029960. MR 0564562. doi:10.1137/1022026 （英语）. Artstein, Zvi. Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points [離散與連續砰砰及面空間，又或：找極值點]. SIAM Review. 1980, 22 (2): 172–185. JSTOR 2029960. MR 0564562. doi:10.1137/1022026 （英语）.
Mas-Colell (1978，第210頁): Mas-Colell, Andreu. A note on the core equivalence theorem: How many blocking coalitions are there? [記核等價定理：有多少個聯盟在阻礙？]. Journal of Mathematical Economics. 1978, 5 (3): 207–215. MR 0514468. doi:10.1016/0304-4068(78)90010-1 （英语）.

archive.org (Global: 6^th place; Chinese: 4^th place)

Starr (1969) Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Guesnerie (1989，第138頁) Guesnerie, Roger. First-best allocation of resources with nonconvexities in production [生產非凸的資源之最優分配]. Cornet, Bernard; Tulkens, Henry (编). Contributions to Operations Research and Economics: The twentieth anniversary of CORE (Papers from the symposium held in Louvain-la-Neuve, January 1987) [運籌學與經濟學投稿：二十周年（[[新魯汶]]1987年舉辦討論會的論文集）]. Cambridge, Mass.: MIT Press. 1989: 99–143. ISBN 0-262-03149-3. MR 1104662 （英语）. 网址－维基内链冲突 (帮助)
Artstein & Vitale (1975，第881–882頁): Artstein, Zvi; Vitale, Richard A. A strong law of large numbers for random compact sets [隨機緊集的強大數定律]. The Annals of Probability. 1975, 3 (5): 879–882. JSTOR 2959130. MR 0385966. Zbl 0313.60012. doi:10.1214/aop/1176996275 （英语）.
Schneider (1993，第xi頁)及Rockafellar (1997，第16頁) Schneider, Rolf. Convex bodies: The Brunn–Minkowski theory [凸體：布倫-閔可夫斯基理論]. Encyclopedia of Mathematics and its Applications 44. Cambridge, UK: Cambridge University Press. 1993: xiv+490. ISBN 0-521-35220-7. MR 1216521 （英语）. Rockafellar, R. Tyrrell. Convex analysis [凸分析]. Princeton Landmarks in Mathematics. Princeton, N.J.: Princeton University Press. 1997: xviii+451. ISBN 0-691-01586-4. MR 1451876 （英语）. 1970年(MR 274683) Princeton Mathematical Series 28的重印版。
Rockafellar (1997，第17頁)及Starr (1997，第78頁) Rockafellar, R. Tyrrell. Convex analysis [凸分析]. Princeton Landmarks in Mathematics. Princeton, N.J.: Princeton University Press. 1997: xviii+451. ISBN 0-691-01586-4. MR 1451876 （英语）. 1970年(MR 274683) Princeton Mathematical Series 28的重印版。 Starr, Ross M. 8 Convex sets, separation theorems, and non-convex sets in R^N (new chapters 22 and 25–26 in (2011) second ed.) [第8章：R^N中的凸集、分離定理、非凸集（及2011年第二版新增的第22、25、26諸章）]. General equilibrium theory: An introduction [一般均衡理論：導論] First. Cambridge, UK: Cambridge University Press. 1997: xxiii+250. ISBN 0-521-56473-5. MR 1462618 （英语）.
Schneider (1993，第2–3頁) Schneider, Rolf. Convex bodies: The Brunn–Minkowski theory [凸體：布倫-閔可夫斯基理論]. Encyclopedia of Mathematics and its Applications 44. Cambridge, UK: Cambridge University Press. 1993: xiv+490. ISBN 0-521-35220-7. MR 1216521 （英语）.
Starr (1969，第35–36頁) Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Schneider (1993，第131頁) Schneider, Rolf. Convex bodies: The Brunn–Minkowski theory [凸體：布倫-閔可夫斯基理論]. Encyclopedia of Mathematics and its Applications 44. Cambridge, UK: Cambridge University Press. 1993: xiv+490. ISBN 0-521-35220-7. MR 1216521 （英语）.
Schneider (1993，第140頁)歸功於Borwein & O'Brien (1978): Borwein, J. M.; O'Brien, R. C. Cancellation characterizes convexity [抵消之事，可以刻畫凸性]. Nanta Mathematica (Nanyang University). 1978, 11: 100–102. ISSN 0077-2739. MR 0510842 （英语）. Schneider, Rolf. Convex bodies: The Brunn–Minkowski theory [凸體：布倫-閔可夫斯基理論]. Encyclopedia of Mathematics and its Applications 44. Cambridge, UK: Cambridge University Press. 1993: xiv+490. ISBN 0-521-35220-7. MR 1216521 （英语）.
Schneider (1993，第129頁) Schneider, Rolf. Convex bodies: The Brunn–Minkowski theory [凸體：布倫-閔可夫斯基理論]. Encyclopedia of Mathematics and its Applications 44. Cambridge, UK: Cambridge University Press. 1993: xiv+490. ISBN 0-521-35220-7. MR 1216521 （英语）.
Starr (1969，第36頁) Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Starr (1969，第37頁) Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Schneider (1993，第129–130頁) Schneider, Rolf. Convex bodies: The Brunn–Minkowski theory [凸體：布倫-閔可夫斯基理論]. Encyclopedia of Mathematics and its Applications 44. Cambridge, UK: Cambridge University Press. 1993: xiv+490. ISBN 0-521-35220-7. MR 1216521 （英语）.
Schneider (1993，第128頁) Schneider, Rolf. Convex bodies: The Brunn–Minkowski theory [凸體：布倫-閔可夫斯基理論]. Encyclopedia of Mathematics and its Applications 44. Cambridge, UK: Cambridge University Press. 1993: xiv+490. ISBN 0-521-35220-7. MR 1216521 （英语）.
Starr (1969，第26頁)：「畢竟，可能覺得車和艇差不多，但多數情況下，半車半艇的組合，既不能駕駛，也不能航行。」（譯文） Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Wold (1943b，第231 and 239–240頁): Wold, Herman. A synthesis of pure demand analysis II [純需求分析綜論二]. Skandinavisk Aktuarietidskrift [斯堪的納維亞精算期刊]. 1943b, 26: 220–263. MR 0011939. doi:10.1080/03461238.1943.10404737.
Wold & Juréen (1953，第146頁): Wold, Herman; Juréen, Lars (in association with Wold). 8 Some further applications of preference fields (pp. 129–148) [八、偏好域的其他應用]. Demand analysis: A study in econometrics [需求分析：計量經濟學研究]. Wiley publications in statistics. New York: John Wiley and Sons, Inc. 1953: xvi+358. MR 0064385 （英语）.
Samuelson (1950，第359–360頁):
會注意到，競爭市場中，不能觀測到無差異曲線凸處（而不是凹）的任何點。此種點被永恆的黑暗遮蔽，除非我等令該消費者壟斷買方，且從非常凸的「預算曲線」上，選取所買的商品。（其沿此曲線，影響所買商品的價格。）在买方垄断的情況，仍可從均衡點觀測到的限制的斜算，推斷該人無差異曲線的斜率。［譯按：此處凸與凹的約定，與本條目相反。］
Samuelson, Paul A. The problem of integrability in utility theory [效用論的可積性問題]. Economica. New Series. November 1950, 17 (68): 355–385. JSTOR 2549499. MR 0043436. doi:10.2307/2549499 （英语）.
「永恆的黑暗」描述彌爾頓所著《失樂園》中的地獄，其卷二第592至594行將地獄的凹陷與塞波尼斯大沼澤（英语：Serbonian Bog）相比：
A gulf profound as that Serbonian Bog
Betwixt Damiata and Mount Casius old,
Where Armies whole have sunk.
彌爾頓對凹陷的描寫，是Arrow & Hahn (1980，第169頁)第7章"Markets with non-convex preferences and production"［非凸偏好與生產的市場］的題辭（英语：epigraph (literature)）。該章講解Starr (1969)的成果。 Arrow, Kenneth J.; Hahn, Frank H. General competitive analysis [一般競爭分析]. Advanced Textbooks in Economics 12 San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6之重印版. Amsterdam: North-Holland. 1980 [1971]. ISBN 0-444-85497-5. MR 0439057 （英语）. Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Farrell, M. J. The Convexity assumption in the theory of competitive markets [競爭市場論的凸性假設]. The Journal of Political Economy. August 1959, 67 (4): 371–391. JSTOR 1825163. doi:10.1086/258197 （英语）.

Farrell, M. J. On Convexity, efficiency, and markets: A Reply [論凸性、效率、市場：回覆]. Journal of Political Economy. October 1961a, 69 (5): 484–489. JSTOR 1828538. doi:10.1086/258541 （英语）.

Farrell, M. J. The Convexity assumption in the theory of competitive markets: Rejoinder [競爭市場論的凸性假設：再回應]. Journal of Political Economy. October 1961b, 69 (5): 493. JSTOR 1828541. doi:10.1086/258544 （英语）.
Koopmans, Tjalling C. Convexity assumptions, allocative efficiency, and competitive equilibrium [凸假設、分配效率、競爭均衡]. The Journal of Political Economy. October 1961, 69 (5): 478–479. JSTOR 1828536. doi:10.1086/258539 （英语）.
Koopmans (1961，第478頁)、Farrell (1959，第390–391頁)、Farrell (1961a，第484頁)、Bator (1961a，第482–483頁)、Rothenberg (1960，第438頁)、Starr (1969，第26頁)評論了Koopmans (1957，第1–126, 尤其 9–16 [1.3 Summation of opportunity sets]、 23–35 [1.6 Convex sets and the price implications of optimality]、 35–37 [1.7 The role of convexity assumptions in the analysis]三節頁)：
Koopmans, Tjalling C. Allocation of resources and the price system [資源分配與價格制度]. Koopmans, Tjalling C (编). Three essays on the state of economic science [三篇論經濟科學現況]. New York: McGraw–Hill Book Company. 1957: 1–126. ISBN 0-07-035337-9 （英语）.
Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Rothenberg (1960，第447頁): Rothenberg, Jerome. Non-convexity, aggregation, and Pareto optimality [非凸性、加總、帕累托最優]. The Journal of Political Economy. October 1960, 68 (5): 435–468. JSTOR 1830308. doi:10.1086/258363 （英语）.

（Rothenberg, Jerome. Comments on non-convexity [評非凸性]. Journal of Political Economy. October 1961, 69 (5): 490–492. JSTOR 1828540. doi:10.1086/258543 （英语）. ）
Aumann (1966，第1–2頁): Aumann, Robert J. Existence of competitive equilibrium in markets with a continuum of traders [市場有連續統多個交易人，則存在競爭均衡]. Econometrica. January 1966, 34 (1): 1–17. JSTOR 1909854. MR 0191623. doi:10.2307/1909854 （英语）. Aumann (1966)用到 Aumann （1964, 1965）的結果：
Aumann, Robert J. Markets with a continuum of traders [連續統多個交易人的市場]. Econometrica. January–April 1964, 32 (1–2): 39–50. JSTOR 1913732. MR 0172689. doi:10.2307/1913732 （英语）.
Aumann, Robert J. Integrals of set-valued functions [集合值函數的積分]. Journal of Mathematical Analysis and Applications. August 1965, 12 (1): 1–12. MR 0185073. doi:10.1016/0022-247X(65)90049-1  （英语）.
Arrow & Hahn (1980，第169–182頁)、Starr (1969，第27–33頁) Arrow, Kenneth J.; Hahn, Frank H. General competitive analysis [一般競爭分析]. Advanced Textbooks in Economics 12 San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6之重印版. Amsterdam: North-Holland. 1980 [1971]. ISBN 0-444-85497-5. MR 0439057 （英语）. Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Guesnerie (1989，第99頁) Guesnerie, Roger. First-best allocation of resources with nonconvexities in production [生產非凸的資源之最優分配]. Cornet, Bernard; Tulkens, Henry (编). Contributions to Operations Research and Economics: The twentieth anniversary of CORE (Papers from the symposium held in Louvain-la-Neuve, January 1987) [運籌學與經濟學投稿：二十周年（[[新魯汶]]1987年舉辦討論會的論文集）]. Cambridge, Mass.: MIT Press. 1989: 99–143. ISBN 0-262-03149-3. MR 1104662 （英语）. 网址－维基内链冲突 (帮助)
Varian (1992，第393–394頁): Varian, Hal R. 21.2 Convexity and size [21.2節：凸性與大小]. Microeconomic Analysis [微觀經濟分析] 3rd. W. W. Norton & Company. 1992. ISBN 978-0-393-95735-8. MR 1036734 （英语）.
Mas-Colell, Whinston & Green (1995，第627–630頁): Mas-Colell, Andreu; Whinston, Michael D.; Green, Jerry R. 17.1 Large economies and nonconvexities [第17.1節：大經濟體與非凸性]. Microeconomic theory [微觀經濟理論]. Oxford University Press. 1995. ISBN 978-0-19-507340-9 （英语）.
Starr (1997，第169頁)，及
Ellickson (1994，第xviii, 306–310, 312, 328–329, 347, and 352頁): Ellickson, Bryan. Competitive equilibrium: Theory and applications [競爭均衡：理論及應用]. Cambridge University Press. 1994. ISBN 978-0-521-31988-1. doi:10.2277/0521319889 （英语）.
Starr, Ross M. 8 Convex sets, separation theorems, and non-convex sets in R^N (new chapters 22 and 25–26 in (2011) second ed.) [第8章：R^N中的凸集、分離定理、非凸集（及2011年第二版新增的第22、25、26諸章）]. General equilibrium theory: An introduction [一般均衡理論：導論] First. Cambridge, UK: Cambridge University Press. 1997: xxiii+250. ISBN 0-521-56473-5. MR 1462618 （英语）.
Salanié (2000，第112–113 and 107–115頁): Salanié, Bernard. 7 Nonconvexities [第7章：非凸性]. Microeconomics of market failures [市場失靈的微觀經濟學] 1998年法文版Microéconomie: Les défaillances du marché (Economica, Paris)的英文翻譯. Cambridge, Mass.: MIT Press. 2000: 107–125. ISBN 0-262-19443-0 （英语）.
Ichiishi (1983，第24–25頁): Ichiishi, Tatsuro. Game theory for economic analysis [經濟分析的賽局理論]. Economic theory, econometrics, and mathematical economics. New York: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]. 1983: x+164. ISBN 0-12-370180-5. MR 0700688 （英语）.
Schneider & Weil (2008，第45頁): Schneider, Rolf; Weil, Wolfgang. Stochastic and integral geometry [隨機與積分幾何]. Probability and its applications. Springer. 2008. ISBN 978-3-540-78858-4. MR 2455326. doi:10.1007/978-3-540-78859-1 （英语）.
Molchanov (2005，第195–198, 218, 232, 237–238 and 407頁): Molchanov, Ilya. 3 Minkowski addition [第3章：閔可夫斯基加法]. Theory of random sets [論隨機集]. Probability and its applications. London: Springer-Verlag London Ltd. 2005: 194–240. ISBN 978-1-84996-949-9. MR 2132405. doi:10.1007/1-84628-150-4 （英语）.
Vind (1964，第168 and 175頁): Vind, Karl. Edgeworth-allocations in an exchange economy with many traders [多交易者交易經濟體中的埃奇沃思分配]. International Economic Review. May 1964, 5 (2): 165–77. JSTOR 2525560. doi:10.2307/2525560 （英语）. 1983年諾貝爾獎得主傑拉德·德布魯有注意維恩（Vind）的論文。Debreu (1991，第4頁)寫道：

凸集的概念（即該集合包含連接其任意兩點的線段），已經多次成為1964年以前經濟理論的核心。引入積分理論研究經濟競爭後，得以新眼光看待此事：若經濟體的每個參與者，對應商品空間的某個任意集合，而又對一族不重要的參與者取平均，則所得的集合必然為凸。[德布魯附註：「此為A. A. 李亞普諾夫的定理的直接推論，參見Vind (1964)。」] 但⋯⋯諸價格函數⋯⋯可以因平均而產生的凸性解釋。商品空間中，對一族不重要參與者加總可以得到凸性，是經濟理論⋯⋯從積分理論得來的觀察。 [刪節後譯文]

Debreu, Gérard. The Mathematization of economic theory [經濟理論的數學化]. The American Economic Review. March 1991, 81 (Presidential address delivered at the 103rd meeting of the American Economic Association, 29 December 1990, Washington, DC): 1–7. JSTOR 2006785 （英语）.

archive.today (Global: 14^th place; Chinese: 18^th place)

Diewert (1982，第552–553頁) Diewert, W. E. 12 Duality approaches to microeconomic theory [第12章：微觀經濟學的對偶進路]. Arrow, Kenneth Joseph; Intriligator, Michael D (编). Handbook of mathematical economics, Volume II [數理經濟學手冊・卷二]. Handbooks in Economics 1. Amsterdam: North-Holland Publishing Co. 1982: 535–599 [2021-08-31]. ISBN 978-0-444-86127-6. MR 0648778. doi:10.1016/S1573-4382(82)02007-4. （原始内容存档于2012-12-10）（英语）.
據Diewert (1982，第552頁)所言，Wold (1943b，第243頁)及Wold & Juréen (1953，第146頁)已於較早前討論過取非凸偏好的凸包。
Diewert, W. E. 12 Duality approaches to microeconomic theory [第12章：微觀經濟學的對偶進路]. Arrow, Kenneth Joseph; Intriligator, Michael D (编). Handbook of mathematical economics, Volume II [數理經濟學手冊・卷二]. Handbooks in Economics 1. Amsterdam: North-Holland Publishing Co. 1982: 535–599 [2021-08-31]. ISBN 978-0-444-86127-6. MR 0648778. doi:10.1016/S1573-4382(82)02007-4. （原始内容存档于2012-12-10）（英语）.

berkeley.edu (Global: 580^th place; Chinese: 642^nd place)

elsa.berkeley.edu

Anderson (2005) Anderson, Robert M. Nonconvex preferences and approximate equilibria [非凸偏好與近似均衡] (PDF). Economics 201B: Economic Theory (Handouts) [經濟學201B: 經濟理論（課堂講義）]. Robert M. Anderson's homepage (Berkeley, Calif.). 14 March 2005: 1–5 [1 January 2011]. （原始内容存档 (PDF)于2012-03-10）.

books.google.com (Global: 3^rd place; Chinese: 8^th place)

Laffont, Jean-Jacques. 3. Nonconvexities [第3章：非凸性]. Fundamentals of public economics [公共經濟學基礎]. MIT Press. 1988: 63–65 [2021-08-30]. ISBN 0-262-12127-1. （原始内容存档于2021-08-30）（英语）.

dictionaryofeconomics.com (Global: 7,008^th place; Chinese: 5,473^rd place)

Starr (2008) Starr, Ross M. Shapley–Folkman theorem [沙普利-福克曼定理]. Durlauf, Steven N.; Blume, Lawrence E (编). The new Palgrave dictionary of economics [新帕爾格雷夫經濟學詞典] Second. Palgrave Macmillan. 2008: 317–318 (1st ed.) [2021-08-31]. ISBN 978-0-333-78676-5. doi:10.1057/9780230226203.1518. （原始内容存档于2017-03-16）（英语）.
Mas-Colell (1987) Mas-Colell, A. Non-convexity [非凸性]. Eatwell, John; Milgate, Murray; Newman, Peter (编). The new Palgrave: A dictionary of economics [新帕爾格雷夫經濟學詞典] first. Palgrave Macmillan. 1987: 653–661 [2021-08-31]. ISBN 9780333786765. doi:10.1057/9780230226203.3173. (Mas-Colell個人主頁的PDF檔). （原始内容存档于2017-11-21）（英语）.

doi.org (Global: 2^nd place; Chinese: 23^rd place)

Starr (1969) Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Starr (2008) Starr, Ross M. Shapley–Folkman theorem [沙普利-福克曼定理]. Durlauf, Steven N.; Blume, Lawrence E (编). The new Palgrave dictionary of economics [新帕爾格雷夫經濟學詞典] Second. Palgrave Macmillan. 2008: 317–318 (1st ed.) [2021-08-31]. ISBN 978-0-333-78676-5. doi:10.1057/9780230226203.1518. （原始内容存档于2017-03-16）（英语）.
Bertsekas (1996，第364–381頁)於p. 374引用Ekeland (1999)，並在p. 381引用Aubin & Ekeland (1976)：
Bertsekas, Dimitri P. 5.6 Large scale separable integer programming problems and the exponential method of multipliers [第5.6節：大規模可分整數規劃問題及乘子指數法]. Constrained optimization and Lagrange multiplier methods [受限優化和拉格朗日乘子法] 1982年Academic Press版的重印. Belmont, Mass.: Athena Scientific. 1996: xiii+395. ISBN 1-886529-04-3. MR 0690767 （英语）.

Bertsekas (1996，第364–381頁)將拉格朗日對偶法用到發電排程（英语：Scheduling (production processes)）上（即機組排程問題（英语：power system simulation）），此種問題有變量限制為整數，所以非凸：

Bertsekas, Dimitri P.; Lauer, Gregory S.; Sandell, Nils R., Jr.; Posbergh, Thomas A. Optimal short-term scheduling of large-scale power systems [大規模電力系統的最優短期排程] (PDF). IEEE Transactions on Automatic Control. January 1983, 28 (1): 1–11 [2 February 2011]. doi:10.1109/tac.1983.1103136. （原始内容存档 (PDF)于2021-09-09）（英语）. Proceedings of 1981 IEEE Conference on Decision and Control, San Diego, CA, December 1981, pp. 432–443.
Ekeland, Ivar. Appendix I: An a priori estimate in convex programming [附錄一：凸規劃的先驗估計]. Ekeland, Ivar; Temam, Roger (编). Convex analysis and variational problems [凸分析與變分問題]. Classics in Applied Mathematics 28 Corrected reprinting of the North-Holland. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). 1999: 357–373 [1976]. ISBN 0-89871-450-8. MR 1727362 （英语）.
Artstein & Vitale (1975，第881–882頁): Artstein, Zvi; Vitale, Richard A. A strong law of large numbers for random compact sets [隨機緊集的強大數定律]. The Annals of Probability. 1975, 3 (5): 879–882. JSTOR 2959130. MR 0385966. Zbl 0313.60012. doi:10.1214/aop/1176996275 （英语）.
Arrow & Hahn (1980，第376頁)、Rockafellar (1997，第10–11頁)、Green & Heller (1981，第37頁) Arrow, Kenneth J.; Hahn, Frank H. General competitive analysis [一般競爭分析]. Advanced Textbooks in Economics 12 San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6之重印版. Amsterdam: North-Holland. 1980 [1971]. ISBN 0-444-85497-5. MR 0439057 （英语）. Rockafellar, R. Tyrrell. Convex analysis [凸分析]. Princeton Landmarks in Mathematics. Princeton, N.J.: Princeton University Press. 1997: xviii+451. ISBN 0-691-01586-4. MR 1451876 （英语）. 1970年(MR 274683) Princeton Mathematical Series 28的重印版。 Green, Jerry; Heller, Walter P. 1 Mathematical analysis and convexity with applications to economics [第1章：數學分析與凸性，及在經濟學之應用]. Arrow, Kenneth Joseph; Intriligator, Michael D (编). Handbook of mathematical economics, Volume I. Handbooks in Economics 1. Amsterdam: North-Holland Publishing Co. 1981: 15–52. ISBN 0-444-86126-2. MR 0634800. doi:10.1016/S1573-4382(81)01005-9 （英语）.
Starr (1969，第35–36頁) Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Starr (1969，第36頁) Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Starr (1969，第37頁) Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Artstein (1980，第180頁) Artstein, Zvi. Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points [離散與連續砰砰及面空間，又或：找極值點]. SIAM Review. 1980, 22 (2): 172–185. JSTOR 2029960. MR 0564562. doi:10.1137/1022026 （英语）.
Starr, Ross M. Approximation of points of convex hull of a sum of sets by points of the sum: An elementary approach [以集合之和的點迫近和集凸包的點：初等進路]. Journal of Economic Theory. 1981, 25 (2): 314–317. MR 0640201. doi:10.1016/0022-0531(81)90010-7 （英语）.
Starr (1969，第26頁)：「畢竟，可能覺得車和艇差不多，但多數情況下，半車半艇的組合，既不能駕駛，也不能航行。」（譯文） Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Hotelling (1935，第74頁): Hotelling, Harold. Demand functions with limited budgets [有限預算的需求函數]. Econometrica. January 1935, 3 (1): 66–78. JSTOR 1907346. doi:10.2307/1907346 （英语）.
Diewert (1982，第552–553頁) Diewert, W. E. 12 Duality approaches to microeconomic theory [第12章：微觀經濟學的對偶進路]. Arrow, Kenneth Joseph; Intriligator, Michael D (编). Handbook of mathematical economics, Volume II [數理經濟學手冊・卷二]. Handbooks in Economics 1. Amsterdam: North-Holland Publishing Co. 1982: 535–599 [2021-08-31]. ISBN 978-0-444-86127-6. MR 0648778. doi:10.1016/S1573-4382(82)02007-4. （原始内容存档于2012-12-10）（英语）.
Wold (1943b，第231 and 239–240頁): Wold, Herman. A synthesis of pure demand analysis II [純需求分析綜論二]. Skandinavisk Aktuarietidskrift [斯堪的納維亞精算期刊]. 1943b, 26: 220–263. MR 0011939. doi:10.1080/03461238.1943.10404737.
Wold & Juréen (1953，第146頁): Wold, Herman; Juréen, Lars (in association with Wold). 8 Some further applications of preference fields (pp. 129–148) [八、偏好域的其他應用]. Demand analysis: A study in econometrics [需求分析：計量經濟學研究]. Wiley publications in statistics. New York: John Wiley and Sons, Inc. 1953: xvi+358. MR 0064385 （英语）.
Samuelson (1950，第359–360頁):
會注意到，競爭市場中，不能觀測到無差異曲線凸處（而不是凹）的任何點。此種點被永恆的黑暗遮蔽，除非我等令該消費者壟斷買方，且從非常凸的「預算曲線」上，選取所買的商品。（其沿此曲線，影響所買商品的價格。）在买方垄断的情況，仍可從均衡點觀測到的限制的斜算，推斷該人無差異曲線的斜率。［譯按：此處凸與凹的約定，與本條目相反。］
Samuelson, Paul A. The problem of integrability in utility theory [效用論的可積性問題]. Economica. New Series. November 1950, 17 (68): 355–385. JSTOR 2549499. MR 0043436. doi:10.2307/2549499 （英语）.
「永恆的黑暗」描述彌爾頓所著《失樂園》中的地獄，其卷二第592至594行將地獄的凹陷與塞波尼斯大沼澤（英语：Serbonian Bog）相比：
A gulf profound as that Serbonian Bog
Betwixt Damiata and Mount Casius old,
Where Armies whole have sunk.
彌爾頓對凹陷的描寫，是Arrow & Hahn (1980，第169頁)第7章"Markets with non-convex preferences and production"［非凸偏好與生產的市場］的題辭（英语：epigraph (literature)）。該章講解Starr (1969)的成果。 Arrow, Kenneth J.; Hahn, Frank H. General competitive analysis [一般競爭分析]. Advanced Textbooks in Economics 12 San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6之重印版. Amsterdam: North-Holland. 1980 [1971]. ISBN 0-444-85497-5. MR 0439057 （英语）. Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Farrell, M. J. The Convexity assumption in the theory of competitive markets [競爭市場論的凸性假設]. The Journal of Political Economy. August 1959, 67 (4): 371–391. JSTOR 1825163. doi:10.1086/258197 （英语）.

Farrell, M. J. On Convexity, efficiency, and markets: A Reply [論凸性、效率、市場：回覆]. Journal of Political Economy. October 1961a, 69 (5): 484–489. JSTOR 1828538. doi:10.1086/258541 （英语）.

Farrell, M. J. The Convexity assumption in the theory of competitive markets: Rejoinder [競爭市場論的凸性假設：再回應]. Journal of Political Economy. October 1961b, 69 (5): 493. JSTOR 1828541. doi:10.1086/258544 （英语）.
Bator, Francis M. On convexity, efficiency, and markets [論凸性、效率、市場]. The Journal of Political Economy. October 1961a, 69 (5): 480–483. JSTOR 1828537. doi:10.1086/258540 （英语）.

Bator, Francis M. On convexity, efficiency, and markets: Rejoinder [論凸性、效率、市場：再回應]. Journal of Political Economy. October 1961b, 69 (5): 489. JSTOR 1828539. doi:10.1086/258542 （英语）.
Koopmans, Tjalling C. Convexity assumptions, allocative efficiency, and competitive equilibrium [凸假設、分配效率、競爭均衡]. The Journal of Political Economy. October 1961, 69 (5): 478–479. JSTOR 1828536. doi:10.1086/258539 （英语）.
Koopmans (1961，第478頁)、Farrell (1959，第390–391頁)、Farrell (1961a，第484頁)、Bator (1961a，第482–483頁)、Rothenberg (1960，第438頁)、Starr (1969，第26頁)評論了Koopmans (1957，第1–126, 尤其 9–16 [1.3 Summation of opportunity sets]、 23–35 [1.6 Convex sets and the price implications of optimality]、 35–37 [1.7 The role of convexity assumptions in the analysis]三節頁)：
Koopmans, Tjalling C. Allocation of resources and the price system [資源分配與價格制度]. Koopmans, Tjalling C (编). Three essays on the state of economic science [三篇論經濟科學現況]. New York: McGraw–Hill Book Company. 1957: 1–126. ISBN 0-07-035337-9 （英语）.
Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Rothenberg (1960，第447頁): Rothenberg, Jerome. Non-convexity, aggregation, and Pareto optimality [非凸性、加總、帕累托最優]. The Journal of Political Economy. October 1960, 68 (5): 435–468. JSTOR 1830308. doi:10.1086/258363 （英语）.

（Rothenberg, Jerome. Comments on non-convexity [評非凸性]. Journal of Political Economy. October 1961, 69 (5): 490–492. JSTOR 1828540. doi:10.1086/258543 （英语）. ）
Shapley & Shubik (1966，第806頁): Shapley, L. S.; Shubik, M. Quasi-cores in a monetary economy with nonconvex preferences [非凸偏好貨幣經濟中的準核]. Econometrica. October 1966, 34 (4): 805–827 [2021-08-30]. JSTOR 1910101. Zbl 0154.45303. doi:10.2307/1910101. （原始内容存档于2017-09-24）（英语）.
Aumann (1966，第1–2頁): Aumann, Robert J. Existence of competitive equilibrium in markets with a continuum of traders [市場有連續統多個交易人，則存在競爭均衡]. Econometrica. January 1966, 34 (1): 1–17. JSTOR 1909854. MR 0191623. doi:10.2307/1909854 （英语）. Aumann (1966)用到 Aumann （1964, 1965）的結果：
Aumann, Robert J. Markets with a continuum of traders [連續統多個交易人的市場]. Econometrica. January–April 1964, 32 (1–2): 39–50. JSTOR 1913732. MR 0172689. doi:10.2307/1913732 （英语）.
Aumann, Robert J. Integrals of set-valued functions [集合值函數的積分]. Journal of Mathematical Analysis and Applications. August 1965, 12 (1): 1–12. MR 0185073. doi:10.1016/0022-247X(65)90049-1  （英语）.
據Diewert (1982，第552頁)所言，Wold (1943b，第243頁)及Wold & Juréen (1953，第146頁)已於較早前討論過取非凸偏好的凸包。
Diewert, W. E. 12 Duality approaches to microeconomic theory [第12章：微觀經濟學的對偶進路]. Arrow, Kenneth Joseph; Intriligator, Michael D (编). Handbook of mathematical economics, Volume II [數理經濟學手冊・卷二]. Handbooks in Economics 1. Amsterdam: North-Holland Publishing Co. 1982: 535–599 [2021-08-31]. ISBN 978-0-444-86127-6. MR 0648778. doi:10.1016/S1573-4382(82)02007-4. （原始内容存档于2012-12-10）（英语）.
Starr & Stinchcombe (1999，第217–218頁): Starr, R. M.; Stinchcombe, M. B. Exchange in a network of trading posts. Chichilnisky, Graciela (编). Markets, information and uncertainty: Essays in economic theory in honor of Kenneth J. Arrow [市場、資訊、不確定性：致敬肯尼斯·阿罗的經濟理論論文]. Cambridge, UK: Cambridge University Press. 1999: 217–234. ISBN 978-0-521-08288-4. doi:10.2277/0521553555 （英语）.
Arrow & Hahn (1980，第169–182頁)、Starr (1969，第27–33頁) Arrow, Kenneth J.; Hahn, Frank H. General competitive analysis [一般競爭分析]. Advanced Textbooks in Economics 12 San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6之重印版. Amsterdam: North-Holland. 1980 [1971]. ISBN 0-444-85497-5. MR 0439057 （英语）. Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Green & Heller (1981，第44頁) Green, Jerry; Heller, Walter P. 1 Mathematical analysis and convexity with applications to economics [第1章：數學分析與凸性，及在經濟學之應用]. Arrow, Kenneth Joseph; Intriligator, Michael D (编). Handbook of mathematical economics, Volume I. Handbooks in Economics 1. Amsterdam: North-Holland Publishing Co. 1981: 15–52. ISBN 0-444-86126-2. MR 0634800. doi:10.1016/S1573-4382(81)01005-9 （英语）.
Mas-Colell (1987) Mas-Colell, A. Non-convexity [非凸性]. Eatwell, John; Milgate, Murray; Newman, Peter (编). The new Palgrave: A dictionary of economics [新帕爾格雷夫經濟學詞典] first. Palgrave Macmillan. 1987: 653–661 [2021-08-31]. ISBN 9780333786765. doi:10.1057/9780230226203.3173. (Mas-Colell個人主頁的PDF檔). （原始内容存档于2017-11-21）（英语）.
Starr (1997，第169頁)，及
Ellickson (1994，第xviii, 306–310, 312, 328–329, 347, and 352頁): Ellickson, Bryan. Competitive equilibrium: Theory and applications [競爭均衡：理論及應用]. Cambridge University Press. 1994. ISBN 978-0-521-31988-1. doi:10.2277/0521319889 （英语）.
Starr, Ross M. 8 Convex sets, separation theorems, and non-convex sets in R^N (new chapters 22 and 25–26 in (2011) second ed.) [第8章：R^N中的凸集、分離定理、非凸集（及2011年第二版新增的第22、25、26諸章）]. General equilibrium theory: An introduction [一般均衡理論：導論] First. Cambridge, UK: Cambridge University Press. 1997: xxiii+250. ISBN 0-521-56473-5. MR 1462618 （英语）.
Aubin & Ekeland (1976，第226, 233, 235, 238, and 241頁): Aubin, J. P.; Ekeland, I. Estimates of the duality gap in nonconvex optimization [估計非凸優化的對偶間隙]. Mathematics of Operations Research. 1976, 1 (3): 225–245. JSTOR 3689565. MR 0449695. doi:10.1287/moor.1.3.225 （英语）.
Aubin & Ekeland (1976)及Ekeland (1999，第362–364頁)也考慮非凸最小值問題的閉凸包，即對原問題的蓋圖（英语：epigraph (mathematics)）取閉凸包所得的新問題。Di Guglielmo推廣到研究非凸多目標優化問題的擬凸閉包，即對目標函數的下水平集取凸閉包所得的問題：
Di Guglielmo (1977，第287–288頁): Di Guglielmo, F. Nonconvex duality in multiobjective optimization [多目標優化的非凸對偶]. Mathematics of Operations Research. 1977, 2 (3): 285–291. JSTOR 3689518. MR 0484418. doi:10.1287/moor.2.3.285 （英语）.

Ekeland, Ivar. Appendix I: An a priori estimate in convex programming [附錄一：凸規劃的先驗估計]. Ekeland, Ivar; Temam, Roger (编). Convex analysis and variational problems [凸分析與變分問題]. Classics in Applied Mathematics 28 Corrected reprinting of the North-Holland. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). 1999: 357–373 [1976]. ISBN 0-89871-450-8. MR 1727362 （英语）.
Schneider & Weil (2008，第45頁): Schneider, Rolf; Weil, Wolfgang. Stochastic and integral geometry [隨機與積分幾何]. Probability and its applications. Springer. 2008. ISBN 978-3-540-78858-4. MR 2455326. doi:10.1007/978-3-540-78859-1 （英语）.
Cassels (1975，第433–434頁): Cassels, J. W. S. Measures of the non-convexity of sets and the Shapley–Folkman–Starr theorem [集合的非凸度與沙普利-福克曼-斯塔定理]. Mathematical Proceedings of the Cambridge Philosophical Society. 1975, 78 (3): 433–436. MR 0385711. doi:10.1017/S0305004100051884 （英语）.
Molchanov (2005，第195–198, 218, 232, 237–238 and 407頁): Molchanov, Ilya. 3 Minkowski addition [第3章：閔可夫斯基加法]. Theory of random sets [論隨機集]. Probability and its applications. London: Springer-Verlag London Ltd. 2005: 194–240. ISBN 978-1-84996-949-9. MR 2132405. doi:10.1007/1-84628-150-4 （英语）.
Puri & Ralescu (1985，第154–155頁): Puri, Madan L.; Ralescu, Dan A. Limit theorems for random compact sets in Banach space [巴拿赫空間隨機緊集的極限定理]. Mathematical Proceedings of the Cambridge Philosophical Society. 1985, 97 (1): 151–158. Bibcode:1985MPCPS..97..151P. MR 0764504. doi:10.1017/S0305004100062691 （英语）.
Weil (1982，第203, and 205–206頁): Weil, Wolfgang. An application of the central limit theorem for Banach-space–valued random variables to the theory of random sets [取值於巴拿赫空間隨機變量的中央極限定理，應用於隨機集理論]. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete [概率論與相關領域期刊]. 1982, 60 (2): 203–208. MR 0663901. doi:10.1007/BF00531823 （英语）.
Cerf (1999，第243–244頁): Cerf, Raphaël. Large deviations for sums of i.i.d. random compact sets [獨立同分佈隨機緊集之和的大離差]. Proceedings of the American Mathematical Society. 1999, 127 (8): 2431–2436. MR 1487361. doi:10.1090/S0002-9939-99-04788-7  （英语）. 瑟夫（Cerf）用到 Puri & Ralescu (1985，第154–155頁)應用沙普利-福克曼引理的結果。
Ruzsa (1997，第345頁): Ruzsa, Imre Z. The Brunn–Minkowski inequality and nonconvex sets [布倫-閔可夫斯基不等式與非凸集]. Geometriae Dedicata. 1997, 67 (3): 337–348. MR 1475877. doi:10.1023/A:1004958110076 （英语）.
Tardella (1990，第478–479頁): Tardella, Fabio. A new proof of the Lyapunov convexity theorem [李亞普諾夫凸性定理的新證]. SIAM Journal on Control and Optimization. 1990, 28 (2): 478–481. MR 1040471. doi:10.1137/0328026 （英语）.
Vind (1964，第168 and 175頁): Vind, Karl. Edgeworth-allocations in an exchange economy with many traders [多交易者交易經濟體中的埃奇沃思分配]. International Economic Review. May 1964, 5 (2): 165–77. JSTOR 2525560. doi:10.2307/2525560 （英语）. 1983年諾貝爾獎得主傑拉德·德布魯有注意維恩（Vind）的論文。Debreu (1991，第4頁)寫道：

凸集的概念（即該集合包含連接其任意兩點的線段），已經多次成為1964年以前經濟理論的核心。引入積分理論研究經濟競爭後，得以新眼光看待此事：若經濟體的每個參與者，對應商品空間的某個任意集合，而又對一族不重要的參與者取平均，則所得的集合必然為凸。[德布魯附註：「此為A. A. 李亞普諾夫的定理的直接推論，參見Vind (1964)。」] 但⋯⋯諸價格函數⋯⋯可以因平均而產生的凸性解釋。商品空間中，對一族不重要參與者加總可以得到凸性，是經濟理論⋯⋯從積分理論得來的觀察。 [刪節後譯文]

Debreu, Gérard. The Mathematization of economic theory [經濟理論的數學化]. The American Economic Review. March 1991, 81 (Presidential address delivered at the 103rd meeting of the American Economic Association, 29 December 1990, Washington, DC): 1–7. JSTOR 2006785 （英语）.
Artstein (1980，第172–183頁) Artstein (1980)在致敬2008年諾貝爾經濟學獎得主羅伯特·奧曼的論文集重新出版：Artstein, Zvi. 22 Discrete and continuous bang–bang and facial spaces or: Look for the extreme points [第22篇：離散與連續砰砰及面空間，又或：找極值點]. Hart, Sergiu; Neyman, Abraham (编). Game and economic theory: Selected contributions in honor of Robert J. Aumann [賽局與經濟理論：致敬羅伯特·J. 奧曼的文選]. Ann Arbor, Mich.: University of Michigan Press. 1995: 449–462. ISBN 0-472-10673-2. （原始内容存档于24 May 2011）（英语）. Artstein, Zvi. Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points [離散與連續砰砰及面空間，又或：找極值點]. SIAM Review. 1980, 22 (2): 172–185. JSTOR 2029960. MR 0564562. doi:10.1137/1022026 （英语）. Artstein, Zvi. Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points [離散與連續砰砰及面空間，又或：找極值點]. SIAM Review. 1980, 22 (2): 172–185. JSTOR 2029960. MR 0564562. doi:10.1137/1022026 （英语）.
Mas-Colell (1978，第210頁): Mas-Colell, Andreu. A note on the core equivalence theorem: How many blocking coalitions are there? [記核等價定理：有多少個聯盟在阻礙？]. Journal of Mathematical Economics. 1978, 5 (3): 207–215. MR 0514468. doi:10.1016/0304-4068(78)90010-1 （英语）.

dtic.mil (Global: 833^rd place; Chinese: 901^st place)

Shapley & Shubik (1966，第806頁): Shapley, L. S.; Shubik, M. Quasi-cores in a monetary economy with nonconvex preferences [非凸偏好貨幣經濟中的準核]. Econometrica. October 1966, 34 (4): 805–827 [2021-08-30]. JSTOR 1910101. Zbl 0154.45303. doi:10.2307/1910101. （原始内容存档于2017-09-24）（英语）.

harvard.edu (Global: 18^th place; Chinese: 57^th place)

ui.adsabs.harvard.edu

Puri & Ralescu (1985，第154–155頁): Puri, Madan L.; Ralescu, Dan A. Limit theorems for random compact sets in Banach space [巴拿赫空間隨機緊集的極限定理]. Mathematical Proceedings of the Cambridge Philosophical Society. 1985, 97 (1): 151–158. Bibcode:1985MPCPS..97..151P. MR 0764504. doi:10.1017/S0305004100062691 （英语）.

jstor.org (Global: 26^th place; Chinese: 113^th place)

Starr (1969) Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Artstein & Vitale (1975，第881–882頁): Artstein, Zvi; Vitale, Richard A. A strong law of large numbers for random compact sets [隨機緊集的強大數定律]. The Annals of Probability. 1975, 3 (5): 879–882. JSTOR 2959130. MR 0385966. Zbl 0313.60012. doi:10.1214/aop/1176996275 （英语）.
Starr (1969，第35–36頁) Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Starr (1969，第36頁) Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Starr (1969，第37頁) Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Artstein (1980，第180頁) Artstein, Zvi. Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points [離散與連續砰砰及面空間，又或：找極值點]. SIAM Review. 1980, 22 (2): 172–185. JSTOR 2029960. MR 0564562. doi:10.1137/1022026 （英语）.
Starr (1969，第26頁)：「畢竟，可能覺得車和艇差不多，但多數情況下，半車半艇的組合，既不能駕駛，也不能航行。」（譯文） Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Hotelling (1935，第74頁): Hotelling, Harold. Demand functions with limited budgets [有限預算的需求函數]. Econometrica. January 1935, 3 (1): 66–78. JSTOR 1907346. doi:10.2307/1907346 （英语）.
Samuelson (1950，第359–360頁):
會注意到，競爭市場中，不能觀測到無差異曲線凸處（而不是凹）的任何點。此種點被永恆的黑暗遮蔽，除非我等令該消費者壟斷買方，且從非常凸的「預算曲線」上，選取所買的商品。（其沿此曲線，影響所買商品的價格。）在买方垄断的情況，仍可從均衡點觀測到的限制的斜算，推斷該人無差異曲線的斜率。［譯按：此處凸與凹的約定，與本條目相反。］
Samuelson, Paul A. The problem of integrability in utility theory [效用論的可積性問題]. Economica. New Series. November 1950, 17 (68): 355–385. JSTOR 2549499. MR 0043436. doi:10.2307/2549499 （英语）.
「永恆的黑暗」描述彌爾頓所著《失樂園》中的地獄，其卷二第592至594行將地獄的凹陷與塞波尼斯大沼澤（英语：Serbonian Bog）相比：
A gulf profound as that Serbonian Bog
Betwixt Damiata and Mount Casius old,
Where Armies whole have sunk.
彌爾頓對凹陷的描寫，是Arrow & Hahn (1980，第169頁)第7章"Markets with non-convex preferences and production"［非凸偏好與生產的市場］的題辭（英语：epigraph (literature)）。該章講解Starr (1969)的成果。 Arrow, Kenneth J.; Hahn, Frank H. General competitive analysis [一般競爭分析]. Advanced Textbooks in Economics 12 San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6之重印版. Amsterdam: North-Holland. 1980 [1971]. ISBN 0-444-85497-5. MR 0439057 （英语）. Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Farrell, M. J. The Convexity assumption in the theory of competitive markets [競爭市場論的凸性假設]. The Journal of Political Economy. August 1959, 67 (4): 371–391. JSTOR 1825163. doi:10.1086/258197 （英语）.

Farrell, M. J. On Convexity, efficiency, and markets: A Reply [論凸性、效率、市場：回覆]. Journal of Political Economy. October 1961a, 69 (5): 484–489. JSTOR 1828538. doi:10.1086/258541 （英语）.

Farrell, M. J. The Convexity assumption in the theory of competitive markets: Rejoinder [競爭市場論的凸性假設：再回應]. Journal of Political Economy. October 1961b, 69 (5): 493. JSTOR 1828541. doi:10.1086/258544 （英语）.
Bator, Francis M. On convexity, efficiency, and markets [論凸性、效率、市場]. The Journal of Political Economy. October 1961a, 69 (5): 480–483. JSTOR 1828537. doi:10.1086/258540 （英语）.

Bator, Francis M. On convexity, efficiency, and markets: Rejoinder [論凸性、效率、市場：再回應]. Journal of Political Economy. October 1961b, 69 (5): 489. JSTOR 1828539. doi:10.1086/258542 （英语）.
Koopmans, Tjalling C. Convexity assumptions, allocative efficiency, and competitive equilibrium [凸假設、分配效率、競爭均衡]. The Journal of Political Economy. October 1961, 69 (5): 478–479. JSTOR 1828536. doi:10.1086/258539 （英语）.
Koopmans (1961，第478頁)、Farrell (1959，第390–391頁)、Farrell (1961a，第484頁)、Bator (1961a，第482–483頁)、Rothenberg (1960，第438頁)、Starr (1969，第26頁)評論了Koopmans (1957，第1–126, 尤其 9–16 [1.3 Summation of opportunity sets]、 23–35 [1.6 Convex sets and the price implications of optimality]、 35–37 [1.7 The role of convexity assumptions in the analysis]三節頁)：
Koopmans, Tjalling C. Allocation of resources and the price system [資源分配與價格制度]. Koopmans, Tjalling C (编). Three essays on the state of economic science [三篇論經濟科學現況]. New York: McGraw–Hill Book Company. 1957: 1–126. ISBN 0-07-035337-9 （英语）.
Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Rothenberg (1960，第447頁): Rothenberg, Jerome. Non-convexity, aggregation, and Pareto optimality [非凸性、加總、帕累托最優]. The Journal of Political Economy. October 1960, 68 (5): 435–468. JSTOR 1830308. doi:10.1086/258363 （英语）.

（Rothenberg, Jerome. Comments on non-convexity [評非凸性]. Journal of Political Economy. October 1961, 69 (5): 490–492. JSTOR 1828540. doi:10.1086/258543 （英语）. ）
Shapley & Shubik (1966，第806頁): Shapley, L. S.; Shubik, M. Quasi-cores in a monetary economy with nonconvex preferences [非凸偏好貨幣經濟中的準核]. Econometrica. October 1966, 34 (4): 805–827 [2021-08-30]. JSTOR 1910101. Zbl 0154.45303. doi:10.2307/1910101. （原始内容存档于2017-09-24）（英语）.
Aumann (1966，第1–2頁): Aumann, Robert J. Existence of competitive equilibrium in markets with a continuum of traders [市場有連續統多個交易人，則存在競爭均衡]. Econometrica. January 1966, 34 (1): 1–17. JSTOR 1909854. MR 0191623. doi:10.2307/1909854 （英语）. Aumann (1966)用到 Aumann （1964, 1965）的結果：
Aumann, Robert J. Markets with a continuum of traders [連續統多個交易人的市場]. Econometrica. January–April 1964, 32 (1–2): 39–50. JSTOR 1913732. MR 0172689. doi:10.2307/1913732 （英语）.
Aumann, Robert J. Integrals of set-valued functions [集合值函數的積分]. Journal of Mathematical Analysis and Applications. August 1965, 12 (1): 1–12. MR 0185073. doi:10.1016/0022-247X(65)90049-1  （英语）.
Arrow & Hahn (1980，第169–182頁)、Starr (1969，第27–33頁) Arrow, Kenneth J.; Hahn, Frank H. General competitive analysis [一般競爭分析]. Advanced Textbooks in Economics 12 San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6之重印版. Amsterdam: North-Holland. 1980 [1971]. ISBN 0-444-85497-5. MR 0439057 （英语）. Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Aubin & Ekeland (1976，第226, 233, 235, 238, and 241頁): Aubin, J. P.; Ekeland, I. Estimates of the duality gap in nonconvex optimization [估計非凸優化的對偶間隙]. Mathematics of Operations Research. 1976, 1 (3): 225–245. JSTOR 3689565. MR 0449695. doi:10.1287/moor.1.3.225 （英语）.
Aubin & Ekeland (1976)及Ekeland (1999，第362–364頁)也考慮非凸最小值問題的閉凸包，即對原問題的蓋圖（英语：epigraph (mathematics)）取閉凸包所得的新問題。Di Guglielmo推廣到研究非凸多目標優化問題的擬凸閉包，即對目標函數的下水平集取凸閉包所得的問題：
Di Guglielmo (1977，第287–288頁): Di Guglielmo, F. Nonconvex duality in multiobjective optimization [多目標優化的非凸對偶]. Mathematics of Operations Research. 1977, 2 (3): 285–291. JSTOR 3689518. MR 0484418. doi:10.1287/moor.2.3.285 （英语）.

Ekeland, Ivar. Appendix I: An a priori estimate in convex programming [附錄一：凸規劃的先驗估計]. Ekeland, Ivar; Temam, Roger (编). Convex analysis and variational problems [凸分析與變分問題]. Classics in Applied Mathematics 28 Corrected reprinting of the North-Holland. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). 1999: 357–373 [1976]. ISBN 0-89871-450-8. MR 1727362 （英语）.
Vind (1964，第168 and 175頁): Vind, Karl. Edgeworth-allocations in an exchange economy with many traders [多交易者交易經濟體中的埃奇沃思分配]. International Economic Review. May 1964, 5 (2): 165–77. JSTOR 2525560. doi:10.2307/2525560 （英语）. 1983年諾貝爾獎得主傑拉德·德布魯有注意維恩（Vind）的論文。Debreu (1991，第4頁)寫道：

凸集的概念（即該集合包含連接其任意兩點的線段），已經多次成為1964年以前經濟理論的核心。引入積分理論研究經濟競爭後，得以新眼光看待此事：若經濟體的每個參與者，對應商品空間的某個任意集合，而又對一族不重要的參與者取平均，則所得的集合必然為凸。[德布魯附註：「此為A. A. 李亞普諾夫的定理的直接推論，參見Vind (1964)。」] 但⋯⋯諸價格函數⋯⋯可以因平均而產生的凸性解釋。商品空間中，對一族不重要參與者加總可以得到凸性，是經濟理論⋯⋯從積分理論得來的觀察。 [刪節後譯文]

Debreu, Gérard. The Mathematization of economic theory [經濟理論的數學化]. The American Economic Review. March 1991, 81 (Presidential address delivered at the 103rd meeting of the American Economic Association, 29 December 1990, Washington, DC): 1–7. JSTOR 2006785 （英语）.
Artstein (1980，第172–183頁) Artstein (1980)在致敬2008年諾貝爾經濟學獎得主羅伯特·奧曼的論文集重新出版：Artstein, Zvi. 22 Discrete and continuous bang–bang and facial spaces or: Look for the extreme points [第22篇：離散與連續砰砰及面空間，又或：找極值點]. Hart, Sergiu; Neyman, Abraham (编). Game and economic theory: Selected contributions in honor of Robert J. Aumann [賽局與經濟理論：致敬羅伯特·J. 奧曼的文選]. Ann Arbor, Mich.: University of Michigan Press. 1995: 449–462. ISBN 0-472-10673-2. （原始内容存档于24 May 2011）（英语）. Artstein, Zvi. Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points [離散與連續砰砰及面空間，又或：找極值點]. SIAM Review. 1980, 22 (2): 172–185. JSTOR 2029960. MR 0564562. doi:10.1137/1022026 （英语）. Artstein, Zvi. Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points [離散與連續砰砰及面空間，又或：找極值點]. SIAM Review. 1980, 22 (2): 172–185. JSTOR 2029960. MR 0564562. doi:10.1137/1022026 （英语）.

mathprog.org (Global: low place; Chinese: low place)

Lemaréchal (1973，第38頁): Lemaréchal, Claude. Utilisation de la dualité dans les problémes non convexes [在非凸問題使用對偶] (报告). Domaine de Voluceau, Rocquencourt, Le Chesnay, France: IRIA（現INRIA）, Laboratoire de recherche en informatique et automatique: 41. April 1973 （法语）. |issue=被忽略 (帮助) 勒馬雷沙爾的實驗，日後有下列論文討論：
Aardal (1995，第2–3頁): Aardal, Karen. Optima interview Claude Lemaréchal [Optima訪問克勞德·勒馬雷沙爾] (PDF). Optima: Mathematical Programming Society Newsletter. March 1995, 45: 2–4 [2 February 2011]. （原始内容存档 (PDF)于2021-09-09）（英语）.

Hiriart-Urruty & Lemaréchal (1993，第143–145, 151, 153, and 156頁): Hiriart-Urruty, Jean-Baptiste; Lemaréchal, Claude. XII Abstract duality for practitioners [第十二章：實踐用的抽象對偶性]. Convex analysis and minimization algorithms, Volume II: Advanced theory and bundle methods [凸分析和最小化算法，第二卷：進階理論及束法]. Grundlehren der Mathematischen Wissenschaften [數理科學的基本原理] 306. Berlin: Springer-Verlag. 1993: 136–193 （及pp. 334–335所列的文獻附註）. ISBN 3-540-56852-2. MR 1295240 （英语）.

michaelcarteronline.com (Global: low place; Chinese: low place)

Carter (2001，第93–94頁)，取n = 3。 Carter, Michael. Foundations of mathematical economics [數理經濟學基礎]. Cambridge, Mass.: MIT Press. 2001: xx+649. ISBN 0-262-53192-5. MR 1865841. (作者網站有習題題解). （原始内容存档于15 September 2006）（英语）.
Carter (2001，第93–94, 143, 318–319, 375–377, and 416頁) Carter, Michael. Foundations of mathematical economics [數理經濟學基礎]. Cambridge, Mass.: MIT Press. 2001: xx+649. ISBN 0-262-53192-5. MR 1865841. (作者網站有習題題解). （原始内容存档于15 September 2006）（英语）.

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mitpress.mit.edu

Carter (2001，第93–94頁)，取n = 3。 Carter, Michael. Foundations of mathematical economics [數理經濟學基礎]. Cambridge, Mass.: MIT Press. 2001: xx+649. ISBN 0-262-53192-5. MR 1865841. (作者網站有習題題解). （原始内容存档于15 September 2006）（英语）.
Carter (2001，第93–94, 143, 318–319, 375–377, and 416頁) Carter, Michael. Foundations of mathematical economics [數理經濟學基礎]. Cambridge, Mass.: MIT Press. 2001: xx+649. ISBN 0-262-53192-5. MR 1865841. (作者網站有習題題解). （原始内容存档于15 September 2006）（英语）.

web.mit.edu

Bertsekas (1996，第364–381頁)於p. 374引用Ekeland (1999)，並在p. 381引用Aubin & Ekeland (1976)：
Bertsekas, Dimitri P. 5.6 Large scale separable integer programming problems and the exponential method of multipliers [第5.6節：大規模可分整數規劃問題及乘子指數法]. Constrained optimization and Lagrange multiplier methods [受限優化和拉格朗日乘子法] 1982年Academic Press版的重印. Belmont, Mass.: Athena Scientific. 1996: xiii+395. ISBN 1-886529-04-3. MR 0690767 （英语）.

Bertsekas (1996，第364–381頁)將拉格朗日對偶法用到發電排程（英语：Scheduling (production processes)）上（即機組排程問題（英语：power system simulation）），此種問題有變量限制為整數，所以非凸：

Bertsekas, Dimitri P.; Lauer, Gregory S.; Sandell, Nils R., Jr.; Posbergh, Thomas A. Optimal short-term scheduling of large-scale power systems [大規模電力系統的最優短期排程] (PDF). IEEE Transactions on Automatic Control. January 1983, 28 (1): 1–11 [2 February 2011]. doi:10.1109/tac.1983.1103136. （原始内容存档 (PDF)于2021-09-09）（英语）. Proceedings of 1981 IEEE Conference on Decision and Control, San Diego, CA, December 1981, pp. 432–443.
Ekeland, Ivar. Appendix I: An a priori estimate in convex programming [附錄一：凸規劃的先驗估計]. Ekeland, Ivar; Temam, Roger (编). Convex analysis and variational problems [凸分析與變分問題]. Classics in Applied Mathematics 28 Corrected reprinting of the North-Holland. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). 1999: 357–373 [1976]. ISBN 0-89871-450-8. MR 1727362 （英语）.

sciencedirect.com (Global: 149^th place; Chinese: 292^nd place)

Diewert (1982，第552–553頁) Diewert, W. E. 12 Duality approaches to microeconomic theory [第12章：微觀經濟學的對偶進路]. Arrow, Kenneth Joseph; Intriligator, Michael D (编). Handbook of mathematical economics, Volume II [數理經濟學手冊・卷二]. Handbooks in Economics 1. Amsterdam: North-Holland Publishing Co. 1982: 535–599 [2021-08-31]. ISBN 978-0-444-86127-6. MR 0648778. doi:10.1016/S1573-4382(82)02007-4. （原始内容存档于2012-12-10）（英语）.
據Diewert (1982，第552頁)所言，Wold (1943b，第243頁)及Wold & Juréen (1953，第146頁)已於較早前討論過取非凸偏好的凸包。
Diewert, W. E. 12 Duality approaches to microeconomic theory [第12章：微觀經濟學的對偶進路]. Arrow, Kenneth Joseph; Intriligator, Michael D (编). Handbook of mathematical economics, Volume II [數理經濟學手冊・卷二]. Handbooks in Economics 1. Amsterdam: North-Holland Publishing Co. 1982: 535–599 [2021-08-31]. ISBN 978-0-444-86127-6. MR 0648778. doi:10.1016/S1573-4382(82)02007-4. （原始内容存档于2012-12-10）（英语）.

umich.edu (Global: 459^th place; Chinese: 752^nd place)

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Artstein (1980，第172–183頁) Artstein (1980)在致敬2008年諾貝爾經濟學獎得主羅伯特·奧曼的論文集重新出版：Artstein, Zvi. 22 Discrete and continuous bang–bang and facial spaces or: Look for the extreme points [第22篇：離散與連續砰砰及面空間，又或：找極值點]. Hart, Sergiu; Neyman, Abraham (编). Game and economic theory: Selected contributions in honor of Robert J. Aumann [賽局與經濟理論：致敬羅伯特·J. 奧曼的文選]. Ann Arbor, Mich.: University of Michigan Press. 1995: 449–462. ISBN 0-472-10673-2. （原始内容存档于24 May 2011）（英语）. Artstein, Zvi. Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points [離散與連續砰砰及面空間，又或：找極值點]. SIAM Review. 1980, 22 (2): 172–185. JSTOR 2029960. MR 0564562. doi:10.1137/1022026 （英语）. Artstein, Zvi. Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points [離散與連續砰砰及面空間，又或：找極值點]. SIAM Review. 1980, 22 (2): 172–185. JSTOR 2029960. MR 0564562. doi:10.1137/1022026 （英语）.

upf.edu (Global: low place; Chinese: low place)

econ.upf.edu

Mas-Colell (1987) Mas-Colell, A. Non-convexity [非凸性]. Eatwell, John; Milgate, Murray; Newman, Peter (编). The new Palgrave: A dictionary of economics [新帕爾格雷夫經濟學詞典] first. Palgrave Macmillan. 1987: 653–661 [2021-08-31]. ISBN 9780333786765. doi:10.1057/9780230226203.3173. (Mas-Colell個人主頁的PDF檔). （原始内容存档于2017-11-21）（英语）.

web.archive.org (Global: 1^st place; Chinese: 1^st place)

Howe (1979，第1頁) Howe, Roger. On the tendency toward convexity of the vector sum of sets [論集合的向量和趨向凸] (PDF) (报告) 538. New Haven, Conn.: Cowles Foundation for Research in Economics, Yale University. 3 November 1979 [1 January 2011]. （原始内容存档 (PDF)于2011-08-07）（英语）.
Starr (2008) Starr, Ross M. Shapley–Folkman theorem [沙普利-福克曼定理]. Durlauf, Steven N.; Blume, Lawrence E (编). The new Palgrave dictionary of economics [新帕爾格雷夫經濟學詞典] Second. Palgrave Macmillan. 2008: 317–318 (1st ed.) [2021-08-31]. ISBN 978-0-333-78676-5. doi:10.1057/9780230226203.1518. （原始内容存档于2017-03-16）（英语）.
Bertsekas (1996，第364–381頁)於p. 374引用Ekeland (1999)，並在p. 381引用Aubin & Ekeland (1976)：
Bertsekas, Dimitri P. 5.6 Large scale separable integer programming problems and the exponential method of multipliers [第5.6節：大規模可分整數規劃問題及乘子指數法]. Constrained optimization and Lagrange multiplier methods [受限優化和拉格朗日乘子法] 1982年Academic Press版的重印. Belmont, Mass.: Athena Scientific. 1996: xiii+395. ISBN 1-886529-04-3. MR 0690767 （英语）.

Bertsekas (1996，第364–381頁)將拉格朗日對偶法用到發電排程（英语：Scheduling (production processes)）上（即機組排程問題（英语：power system simulation）），此種問題有變量限制為整數，所以非凸：

Bertsekas, Dimitri P.; Lauer, Gregory S.; Sandell, Nils R., Jr.; Posbergh, Thomas A. Optimal short-term scheduling of large-scale power systems [大規模電力系統的最優短期排程] (PDF). IEEE Transactions on Automatic Control. January 1983, 28 (1): 1–11 [2 February 2011]. doi:10.1109/tac.1983.1103136. （原始内容存档 (PDF)于2021-09-09）（英语）. Proceedings of 1981 IEEE Conference on Decision and Control, San Diego, CA, December 1981, pp. 432–443.
Ekeland, Ivar. Appendix I: An a priori estimate in convex programming [附錄一：凸規劃的先驗估計]. Ekeland, Ivar; Temam, Roger (编). Convex analysis and variational problems [凸分析與變分問題]. Classics in Applied Mathematics 28 Corrected reprinting of the North-Holland. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). 1999: 357–373 [1976]. ISBN 0-89871-450-8. MR 1727362 （英语）.
Carter (2001，第93–94頁)，取n = 3。 Carter, Michael. Foundations of mathematical economics [數理經濟學基礎]. Cambridge, Mass.: MIT Press. 2001: xx+649. ISBN 0-262-53192-5. MR 1865841. (作者網站有習題題解). （原始内容存档于15 September 2006）（英语）.
Anderson (2005) Anderson, Robert M. Nonconvex preferences and approximate equilibria [非凸偏好與近似均衡] (PDF). Economics 201B: Economic Theory (Handouts) [經濟學201B: 經濟理論（課堂講義）]. Robert M. Anderson's homepage (Berkeley, Calif.). 14 March 2005: 1–5 [1 January 2011]. （原始内容存档 (PDF)于2012-03-10）.
Shapley & Shubik (1966，第806頁): Shapley, L. S.; Shubik, M. Quasi-cores in a monetary economy with nonconvex preferences [非凸偏好貨幣經濟中的準核]. Econometrica. October 1966, 34 (4): 805–827 [2021-08-30]. JSTOR 1910101. Zbl 0154.45303. doi:10.2307/1910101. （原始内容存档于2017-09-24）（英语）.
Mas-Colell (1987) Mas-Colell, A. Non-convexity [非凸性]. Eatwell, John; Milgate, Murray; Newman, Peter (编). The new Palgrave: A dictionary of economics [新帕爾格雷夫經濟學詞典] first. Palgrave Macmillan. 1987: 653–661 [2021-08-31]. ISBN 9780333786765. doi:10.1057/9780230226203.3173. (Mas-Colell個人主頁的PDF檔). （原始内容存档于2017-11-21）（英语）.
Laffont, Jean-Jacques. 3. Nonconvexities [第3章：非凸性]. Fundamentals of public economics [公共經濟學基礎]. MIT Press. 1988: 63–65 [2021-08-30]. ISBN 0-262-12127-1. （原始内容存档于2021-08-30）（英语）.
Carter (2001，第93–94, 143, 318–319, 375–377, and 416頁) Carter, Michael. Foundations of mathematical economics [數理經濟學基礎]. Cambridge, Mass.: MIT Press. 2001: xx+649. ISBN 0-262-53192-5. MR 1865841. (作者網站有習題題解). （原始内容存档于15 September 2006）（英语）.
Lemaréchal (1973，第38頁): Lemaréchal, Claude. Utilisation de la dualité dans les problémes non convexes [在非凸問題使用對偶] (报告). Domaine de Voluceau, Rocquencourt, Le Chesnay, France: IRIA（現INRIA）, Laboratoire de recherche en informatique et automatique: 41. April 1973 （法语）. |issue=被忽略 (帮助) 勒馬雷沙爾的實驗，日後有下列論文討論：
Aardal (1995，第2–3頁): Aardal, Karen. Optima interview Claude Lemaréchal [Optima訪問克勞德·勒馬雷沙爾] (PDF). Optima: Mathematical Programming Society Newsletter. March 1995, 45: 2–4 [2 February 2011]. （原始内容存档 (PDF)于2021-09-09）（英语）.

Hiriart-Urruty & Lemaréchal (1993，第143–145, 151, 153, and 156頁): Hiriart-Urruty, Jean-Baptiste; Lemaréchal, Claude. XII Abstract duality for practitioners [第十二章：實踐用的抽象對偶性]. Convex analysis and minimization algorithms, Volume II: Advanced theory and bundle methods [凸分析和最小化算法，第二卷：進階理論及束法]. Grundlehren der Mathematischen Wissenschaften [數理科學的基本原理] 306. Berlin: Springer-Verlag. 1993: 136–193 （及pp. 334–335所列的文獻附註）. ISBN 3-540-56852-2. MR 1295240 （英语）.
Artstein (1980，第172–183頁) Artstein (1980)在致敬2008年諾貝爾經濟學獎得主羅伯特·奧曼的論文集重新出版：Artstein, Zvi. 22 Discrete and continuous bang–bang and facial spaces or: Look for the extreme points [第22篇：離散與連續砰砰及面空間，又或：找極值點]. Hart, Sergiu; Neyman, Abraham (编). Game and economic theory: Selected contributions in honor of Robert J. Aumann [賽局與經濟理論：致敬羅伯特·J. 奧曼的文選]. Ann Arbor, Mich.: University of Michigan Press. 1995: 449–462. ISBN 0-472-10673-2. （原始内容存档于24 May 2011）（英语）. Artstein, Zvi. Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points [離散與連續砰砰及面空間，又或：找極值點]. SIAM Review. 1980, 22 (2): 172–185. JSTOR 2029960. MR 0564562. doi:10.1137/1022026 （英语）. Artstein, Zvi. Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points [離散與連續砰砰及面空間，又或：找極值點]. SIAM Review. 1980, 22 (2): 172–185. JSTOR 2029960. MR 0564562. doi:10.1137/1022026 （英语）.

wikipedia.org (Global: low place; Chinese: low place)

en.wikipedia.org

Bertsekas (1996，第364–381頁)於p. 374引用Ekeland (1999)，並在p. 381引用Aubin & Ekeland (1976)：
Bertsekas, Dimitri P. 5.6 Large scale separable integer programming problems and the exponential method of multipliers [第5.6節：大規模可分整數規劃問題及乘子指數法]. Constrained optimization and Lagrange multiplier methods [受限優化和拉格朗日乘子法] 1982年Academic Press版的重印. Belmont, Mass.: Athena Scientific. 1996: xiii+395. ISBN 1-886529-04-3. MR 0690767 （英语）.

Bertsekas (1996，第364–381頁)將拉格朗日對偶法用到發電排程（英语：Scheduling (production processes)）上（即機組排程問題（英语：power system simulation）），此種問題有變量限制為整數，所以非凸：

Bertsekas, Dimitri P.; Lauer, Gregory S.; Sandell, Nils R., Jr.; Posbergh, Thomas A. Optimal short-term scheduling of large-scale power systems [大規模電力系統的最優短期排程] (PDF). IEEE Transactions on Automatic Control. January 1983, 28 (1): 1–11 [2 February 2011]. doi:10.1109/tac.1983.1103136. （原始内容存档 (PDF)于2021-09-09）（英语）. Proceedings of 1981 IEEE Conference on Decision and Control, San Diego, CA, December 1981, pp. 432–443.
Ekeland, Ivar. Appendix I: An a priori estimate in convex programming [附錄一：凸規劃的先驗估計]. Ekeland, Ivar; Temam, Roger (编). Convex analysis and variational problems [凸分析與變分問題]. Classics in Applied Mathematics 28 Corrected reprinting of the North-Holland. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). 1999: 357–373 [1976]. ISBN 0-89871-450-8. MR 1727362 （英语）.
Samuelson (1950，第359–360頁):
會注意到，競爭市場中，不能觀測到無差異曲線凸處（而不是凹）的任何點。此種點被永恆的黑暗遮蔽，除非我等令該消費者壟斷買方，且從非常凸的「預算曲線」上，選取所買的商品。（其沿此曲線，影響所買商品的價格。）在买方垄断的情況，仍可從均衡點觀測到的限制的斜算，推斷該人無差異曲線的斜率。［譯按：此處凸與凹的約定，與本條目相反。］
Samuelson, Paul A. The problem of integrability in utility theory [效用論的可積性問題]. Economica. New Series. November 1950, 17 (68): 355–385. JSTOR 2549499. MR 0043436. doi:10.2307/2549499 （英语）.
「永恆的黑暗」描述彌爾頓所著《失樂園》中的地獄，其卷二第592至594行將地獄的凹陷與塞波尼斯大沼澤（英语：Serbonian Bog）相比：
A gulf profound as that Serbonian Bog
Betwixt Damiata and Mount Casius old,
Where Armies whole have sunk.
彌爾頓對凹陷的描寫，是Arrow & Hahn (1980，第169頁)第7章"Markets with non-convex preferences and production"［非凸偏好與生產的市場］的題辭（英语：epigraph (literature)）。該章講解Starr (1969)的成果。 Arrow, Kenneth J.; Hahn, Frank H. General competitive analysis [一般競爭分析]. Advanced Textbooks in Economics 12 San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6之重印版. Amsterdam: North-Holland. 1980 [1971]. ISBN 0-444-85497-5. MR 0439057 （英语）. Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.
Aubin & Ekeland (1976，第226, 233, 235, 238, and 241頁): Aubin, J. P.; Ekeland, I. Estimates of the duality gap in nonconvex optimization [估計非凸優化的對偶間隙]. Mathematics of Operations Research. 1976, 1 (3): 225–245. JSTOR 3689565. MR 0449695. doi:10.1287/moor.1.3.225 （英语）.
Aubin & Ekeland (1976)及Ekeland (1999，第362–364頁)也考慮非凸最小值問題的閉凸包，即對原問題的蓋圖（英语：epigraph (mathematics)）取閉凸包所得的新問題。Di Guglielmo推廣到研究非凸多目標優化問題的擬凸閉包，即對目標函數的下水平集取凸閉包所得的問題：
Di Guglielmo (1977，第287–288頁): Di Guglielmo, F. Nonconvex duality in multiobjective optimization [多目標優化的非凸對偶]. Mathematics of Operations Research. 1977, 2 (3): 285–291. JSTOR 3689518. MR 0484418. doi:10.1287/moor.2.3.285 （英语）.

Ekeland, Ivar. Appendix I: An a priori estimate in convex programming [附錄一：凸規劃的先驗估計]. Ekeland, Ivar; Temam, Roger (编). Convex analysis and variational problems [凸分析與變分問題]. Classics in Applied Mathematics 28 Corrected reprinting of the North-Holland. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). 1999: 357–373 [1976]. ISBN 0-89871-450-8. MR 1727362 （英语）.

wikisource.org (Global: 27^th place; Chinese: 30^th place)

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Samuelson (1950，第359–360頁):
會注意到，競爭市場中，不能觀測到無差異曲線凸處（而不是凹）的任何點。此種點被永恆的黑暗遮蔽，除非我等令該消費者壟斷買方，且從非常凸的「預算曲線」上，選取所買的商品。（其沿此曲線，影響所買商品的價格。）在买方垄断的情況，仍可從均衡點觀測到的限制的斜算，推斷該人無差異曲線的斜率。［譯按：此處凸與凹的約定，與本條目相反。］
Samuelson, Paul A. The problem of integrability in utility theory [效用論的可積性問題]. Economica. New Series. November 1950, 17 (68): 355–385. JSTOR 2549499. MR 0043436. doi:10.2307/2549499 （英语）.
「永恆的黑暗」描述彌爾頓所著《失樂園》中的地獄，其卷二第592至594行將地獄的凹陷與塞波尼斯大沼澤（英语：Serbonian Bog）相比：
A gulf profound as that Serbonian Bog
Betwixt Damiata and Mount Casius old,
Where Armies whole have sunk.
彌爾頓對凹陷的描寫，是Arrow & Hahn (1980，第169頁)第7章"Markets with non-convex preferences and production"［非凸偏好與生產的市場］的題辭（英语：epigraph (literature)）。該章講解Starr (1969)的成果。 Arrow, Kenneth J.; Hahn, Frank H. General competitive analysis [一般競爭分析]. Advanced Textbooks in Economics 12 San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6之重印版. Amsterdam: North-Holland. 1980 [1971]. ISBN 0-444-85497-5. MR 0439057 （英语）. Starr, Ross M. Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37) [有非凸偏好的市場的準均衡（附錄2:沙普利-福克曼定理，pp. 35–37）]. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201 （英语）.

worldcat.org (Global: 5^th place; Chinese: 12^th place)

Schneider (1993，第140頁)歸功於Borwein & O'Brien (1978): Borwein, J. M.; O'Brien, R. C. Cancellation characterizes convexity [抵消之事，可以刻畫凸性]. Nanta Mathematica (Nanyang University). 1978, 11: 100–102. ISSN 0077-2739. MR 0510842 （英语）. Schneider, Rolf. Convex bodies: The Brunn–Minkowski theory [凸體：布倫-閔可夫斯基理論]. Encyclopedia of Mathematics and its Applications 44. Cambridge, UK: Cambridge University Press. 1993: xiv+490. ISBN 0-521-35220-7. MR 1216521 （英语）.
Ekeland, Ivar. Une estimation a priori en programmation non convexe [非凸規劃中的先驗估計]. Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences [（法國）科學院週刊]. Séries A et B. 1974, 279: 149–151. ISSN 0151-0509. MR 0395844 （法语）.

yale.edu (Global: 565^th place; Chinese: 664^th place)

cowles.econ.yale.edu

Howe (1979，第1頁) Howe, Roger. On the tendency toward convexity of the vector sum of sets [論集合的向量和趨向凸] (PDF) (报告) 538. New Haven, Conn.: Cowles Foundation for Research in Economics, Yale University. 3 November 1979 [1 January 2011]. （原始内容存档 (PDF)于2011-08-07）（英语）.

zbmath.org (Global: 1,923^rd place; Chinese: low place)

Artstein & Vitale (1975，第881–882頁): Artstein, Zvi; Vitale, Richard A. A strong law of large numbers for random compact sets [隨機緊集的強大數定律]. The Annals of Probability. 1975, 3 (5): 879–882. JSTOR 2959130. MR 0385966. Zbl 0313.60012. doi:10.1214/aop/1176996275 （英语）.
Shapley & Shubik (1966，第806頁): Shapley, L. S.; Shubik, M. Quasi-cores in a monetary economy with nonconvex preferences [非凸偏好貨幣經濟中的準核]. Econometrica. October 1966, 34 (4): 805–827 [2021-08-30]. JSTOR 1910101. Zbl 0154.45303. doi:10.2307/1910101. （原始内容存档于2017-09-24）（英语）.

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