Λήμμα των Σάπλεϊ-Φόλκμαν (Greek Wikipedia)

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Guesnerie (1989, p. 138) Guesnerie, Roger (1989). «First-best allocation of resources with nonconvexities in production». Στο: Cornet, Bernard· Tulkens, Henry. Contributions to Operations Research and Economics: The twentieth anniversary of CORE (Papers from the symposium held in Louvain-la-Neuve, January 1987). Cambridge, MA: MIT Press. σελίδες 99–143. ISBN 0-262-03149-3. MR 1104662.
(Ekeland 1999, σελίδες 357–359): Published in the first English edition of 1976, Ekeland's appendix proves the Shapley–Folkman lemma, also acknowledging Lemaréchal's experiments on page 373. Ekeland, Ivar (1999) [1976]. «Appendix I: An a priori estimate in convex programming». Στο: Ekeland, Ivar· Temam, Roger. Convex analysis and variational problems. Classics in Applied Mathematics. 28 (Corrected reprinting of the North-Holland έκδοση). Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). σελίδες 357–373. ISBN 0-89871-450-8. MR 1727362.
Bertsekas (1996, pp. 364–381) acknowledging Ekeland (1999) on page 374 and Aubin & Ekeland (1976) on page 381:
Bertsekas, Dimitri P. (1996). «5.6 Large scale separable integer programming problems and the exponential method of multipliers». Constrained optimization and Lagrange multiplier methods (Reprint of (1982) Academic Press έκδοση). Belmont, MA: Athena Scientific. σελίδες xiii+395. ISBN 1-886529-04-3. MR 0690767.

Bertsekas (1996, pp. 364–381) describes an application of Lagrangian dual methods to the scheduling of electrical power plants ("unit commitment problems"), where non-convexity appears because of integer constraints:
Bertsekas, Dimitri P.; Lauer, Gregory S.; Sandell, Nils R., Jr.; Posbergh, Thomas A. (January 1983). «Optimal short-term scheduling of large-scale power systems». IEEE Transactions on Automatic Control AC-28 (Proceedings of 1981 IEEE Conference on Decision and Control, San Diego, CA, December 1981, pp.432–443): 1–11. http://web.mit.edu/dimitrib/www/Unit_Comm.pdf. Ανακτήθηκε στις 2011-02-02.
Ekeland, Ivar (1999) [1976]. «Appendix I: An a priori estimate in convex programming». Στο: Ekeland, Ivar· Temam, Roger. Convex analysis and variational problems. Classics in Applied Mathematics. 28 (Corrected reprinting of the North-Holland έκδοση). Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). σελίδες 357–373. ISBN 0-89871-450-8. MR 1727362.
Carter (2001, p. 94) Carter, Michael (2001). Foundations of mathematical economics. Cambridge, MA: MIT Press. σελίδες xx+649. ISBN 0-262-53192-5. MR 1865841. (Author's website with answers to exercises). Αρχειοθετήθηκε από το πρωτότυπο στις 15 Σεπτεμβρίου 2006. Ανακτήθηκε στις 25 Ιουνίου 2014.
Arrow & Hahn (1980, p. 375) Arrow, Kenneth J.· Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 έκδοση). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057.
Rockafellar (1997, p. 10) Rockafellar, R. Tyrrell (1997). Convex analysis. Princeton Landmarks in Mathematics (Reprint of the 1970 (MR 274683) Princeton Mathematical Series 28 έκδοση). Princeton, NJ: Princeton University Press. σελίδες xviii+451. ISBN 0-691-01586-4. MR 1451876.
Arrow & Hahn (1980, p. 376), Rockafellar (1997, pp. 10–11), and Green & Heller (1981, p. 37) Arrow, Kenneth J.· Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 έκδοση). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057. Rockafellar, R. Tyrrell (1997). Convex analysis. Princeton Landmarks in Mathematics (Reprint of the 1970 (MR 274683) Princeton Mathematical Series 28 έκδοση). Princeton, NJ: Princeton University Press. σελίδες xviii+451. ISBN 0-691-01586-4. MR 1451876. Green, Jerry· Heller, Walter P. (1981). «1 Mathematical analysis and convexity with applications to economics». Στο: Arrow, Kenneth Joseph· Intriligator, Michael D. Handbook of mathematical economics, Volume I. Handbooks in Economics. 1. Amsterdam: North-Holland Publishing Co. σελίδες 15–52. doi:10.1016/S1573-4382(81)01005-9. ISBN 0-444-86126-2. MR 0634800. Αρχειοθετήθηκε από το πρωτότυπο στις 17 Δεκεμβρίου 2012. Ανακτήθηκε στις 25 Ιουνίου 2014.
Arrow & Hahn (1980, p. 385) and Rockafellar (1997, pp. 11–12) Arrow, Kenneth J.· Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 έκδοση). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057. Rockafellar, R. Tyrrell (1997). Convex analysis. Princeton Landmarks in Mathematics (Reprint of the 1970 (MR 274683) Princeton Mathematical Series 28 έκδοση). Princeton, NJ: Princeton University Press. σελίδες xviii+451. ISBN 0-691-01586-4. MR 1451876.
Schneider (1993, p. xi) and Rockafellar (1997, p. 16) Schneider, Rolf (1993). Convex bodies: The Brunn–Minkowski theory. Encyclopedia of Mathematics and its Applications. 44. Cambridge: Cambridge University Press. σελίδες xiv+490. ISBN 0-521-35220-7. MR 1216521. Rockafellar, R. Tyrrell (1997). Convex analysis. Princeton Landmarks in Mathematics (Reprint of the 1970 (MR 274683) Princeton Mathematical Series 28 έκδοση). Princeton, NJ: Princeton University Press. σελίδες xviii+451. ISBN 0-691-01586-4. MR 1451876.
Rockafellar (1997, p. 17) and Starr (1997, p. 78) Rockafellar, R. Tyrrell (1997). Convex analysis. Princeton Landmarks in Mathematics (Reprint of the 1970 (MR 274683) Princeton Mathematical Series 28 έκδοση). Princeton, NJ: Princeton University Press. σελίδες xviii+451. ISBN 0-691-01586-4. MR 1451876.
Schneider (1993, pp. 2–3) Schneider, Rolf (1993). Convex bodies: The Brunn–Minkowski theory. Encyclopedia of Mathematics and its Applications. 44. Cambridge: Cambridge University Press. σελίδες xiv+490. ISBN 0-521-35220-7. MR 1216521.
Arrow & Hahn (1980, p. 387) Arrow, Kenneth J.· Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 έκδοση). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057.
Schneider (1993, p. 131) Schneider, Rolf (1993). Convex bodies: The Brunn–Minkowski theory. Encyclopedia of Mathematics and its Applications. 44. Cambridge: Cambridge University Press. σελίδες xiv+490. ISBN 0-521-35220-7. MR 1216521.
Schneider (1993, p. 140) credits this result to Borwein & O'Brien (1978): Borwein, Jonathan M.; O'Brien, R. C. (1978). «Cancellation characterizes convexity». Nanta Mathematica (Nanyang University) 11: 100–102. ISSN 0077-2739. MR 510842. Schneider, Rolf (1993). Convex bodies: The Brunn–Minkowski theory. Encyclopedia of Mathematics and its Applications. 44. Cambridge: Cambridge University Press. σελίδες xiv+490. ISBN 0-521-35220-7. MR 1216521.
Schneider (1993, p. 129) Schneider, Rolf (1993). Convex bodies: The Brunn–Minkowski theory. Encyclopedia of Mathematics and its Applications. 44. Cambridge: Cambridge University Press. σελίδες xiv+490. ISBN 0-521-35220-7. MR 1216521.
Schneider (1993, pp. 129–130) Schneider, Rolf (1993). Convex bodies: The Brunn–Minkowski theory. Encyclopedia of Mathematics and its Applications. 44. Cambridge: Cambridge University Press. σελίδες xiv+490. ISBN 0-521-35220-7. MR 1216521.
Arrow & Hahn (1980, pp. 392–395) Arrow, Kenneth J.· Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 έκδοση). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057.
Schneider (1993, p. 128) Schneider, Rolf (1993). Convex bodies: The Brunn–Minkowski theory. Encyclopedia of Mathematics and its Applications. 44. Cambridge: Cambridge University Press. σελίδες xiv+490. ISBN 0-521-35220-7. MR 1216521.
Ekeland (1999, pp. 357–359) Ekeland, Ivar (1999) [1976]. «Appendix I: An a priori estimate in convex programming». Στο: Ekeland, Ivar· Temam, Roger. Convex analysis and variational problems. Classics in Applied Mathematics. 28 (Corrected reprinting of the North-Holland έκδοση). Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). σελίδες 357–373. ISBN 0-89871-450-8. MR 1727362.
Starr, Ross M. (1981). «Approximation of points of convex hull of a sum of sets by points of the sum: An elementary approach». Journal of Economic Theory 25 (2): 314–317. doi:10.1016/0022-0531(81)90010-7. MR 640201. Αρχειοθετήθηκε από το πρωτότυπο στις 2013-02-02. https://archive.today/20130202094407/http://www.sciencedirect.com/science/article/pii/0022053181900107. Ανακτήθηκε στις 2014-06-25.
Mas-Colell (1985, pp. 58–61) and Arrow & Hahn (1980, pp. 76–79) Arrow, Kenneth J.· Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 έκδοση). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057.
Arrow & Hahn (1980, pp. 79–81) Arrow, Kenneth J.· Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 έκδοση). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057.
Wold (1943b, pp. 231 and 239–240): Wold, Herman (1943b). «A synthesis of pure demand analysis II». Skandinavisk Aktuarietidskrift [Scandinavian Actuarial Journal] 26: 220–263. MR 11939.
Wold & Juréen (1953, p. 146): Wold, Herman· Juréen, Lars (in association with Wold) (1953). «8 Some further applications of preference fields (pp. 129–148)». Demand analysis: A study in econometrics. Wiley publications in statistics. New York: John Wiley and Sons, Inc. σελίδες xvi+358. MR 0064385.
Samuelson (1950, pp. 359–360):
It will be noted that any point where the indifference curves are convex rather than concave cannot be observed in a competitive market. Such points are shrouded in eternal darkness—unless we make our consumer a monopsonist and let him choose between goods lying on a very convex "budget curve" (along which he is affecting the price of what he buys). In this monopsony case, we could still deduce the slope of the man's indifference curve from the slope of the observed constraint at the equilibrium point.
Samuelson, Paul A. (Νοέμβριος 1950). «The problem of integrability in utility theory». Economica. New Series 17 (68): 355–385. MR 43436. "Eternal darkness" describes the Hell of John Milton's Paradise Lost, whose concavity is compared to the Serbonian Bog in Book II, lines 592–594:
A gulf profound as that Serbonian Bog

Betwixt Damiata and Mount Casius old,

Where Armies whole have sunk.
Milton's description of concavity serves as the literary epigraph prefacing chapter seven of Arrow & Hahn (1971, p. 169), "Markets with non-convex preferences and production", which presents the results of Starr (1969).
Diewert (1982, pp. 552–553) Diewert, W. E. (1982). «12 Duality approaches to microeconomic theory». Στο: Arrow, Kenneth Joseph· Intriligator, Michael D. Handbook of mathematical economics, Volume II. Handbooks in Economics. 1. Amsterdam: North-Holland Publishing Co. σελίδες 535–599. doi:10.1016/S1573-4382(82)02007-4. ISBN 978-0-444-86127-6. MR 0648778. Αρχειοθετήθηκε από το πρωτότυπο στις 10 Δεκεμβρίου 2012. Ανακτήθηκε στις 25 Ιουνίου 2014.
Arrow & Hahn (1980, p. 182) Arrow, Kenneth J.· Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 έκδοση). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057.
Aumann (1966, pp. 1–2): Aumann, Robert J. (January 1966). «Existence of competitive equilibrium in markets with a continuum of traders». Econometrica 34 (1): 1–17. MR 191623. https://archive.org/details/sim_econometrica_1966-01_34_1/page/1. Aumann (1966) uses results from Aumann (1964, 1965): Aumann, Robert J. (Ιανουάριος–Απρίλιος 1964). «Markets with a continuum of traders». Econometrica 32 (1–2): 39–50. MR 172689. Aumann, Robert J. (Αύγουστος 1965). «Integrals of set-valued functions». Journal of Mathematical Analysis and Applications 12 (1): 1–12. doi:10.1016/0022-247X(65)90049-1. MR 185073. Αρχειοθετήθηκε από το πρωτότυπο στις 2013-02-02. https://archive.today/20130202032422/http://www.sciencedirect.com/science/article/pii/0022247X65900491. Ανακτήθηκε στις 2014-06-25.
Taking the convex hull of non-convex preferences had been discussed earlier by Wold (1943b, p. 243) and by Wold & Juréen (1953, p. 146), according to Diewert (1982, p. 552). Diewert, W. E. (1982). «12 Duality approaches to microeconomic theory». Στο: Arrow, Kenneth Joseph· Intriligator, Michael D. Handbook of mathematical economics, Volume II. Handbooks in Economics. 1. Amsterdam: North-Holland Publishing Co. σελίδες 535–599. doi:10.1016/S1573-4382(82)02007-4. ISBN 978-0-444-86127-6. MR 0648778. Αρχειοθετήθηκε από το πρωτότυπο στις 10 Δεκεμβρίου 2012. Ανακτήθηκε στις 25 Ιουνίου 2014.
Arrow & Hahn (1980, pp. 169–182). Starr (1969, pp. 27–33) Arrow, Kenneth J.· Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 έκδοση). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057.
Green & Heller (1981, p. 44) Green, Jerry· Heller, Walter P. (1981). «1 Mathematical analysis and convexity with applications to economics». Στο: Arrow, Kenneth Joseph· Intriligator, Michael D. Handbook of mathematical economics, Volume I. Handbooks in Economics. 1. Amsterdam: North-Holland Publishing Co. σελίδες 15–52. doi:10.1016/S1573-4382(81)01005-9. ISBN 0-444-86126-2. MR 0634800. Αρχειοθετήθηκε από το πρωτότυπο στις 17 Δεκεμβρίου 2012. Ανακτήθηκε στις 25 Ιουνίου 2014.
Guesnerie (1989, pp. 99) Guesnerie, Roger (1989). «First-best allocation of resources with nonconvexities in production». Στο: Cornet, Bernard· Tulkens, Henry. Contributions to Operations Research and Economics: The twentieth anniversary of CORE (Papers from the symposium held in Louvain-la-Neuve, January 1987). Cambridge, MA: MIT Press. σελίδες 99–143. ISBN 0-262-03149-3. MR 1104662.
Varian (1992, pp. 393–394): Varian, Hal R. (1992). «21.2 Convexity and size». Microeconomic Analysis (3η έκδοση). W. W. Norton & Company. ISBN 978-0-393-95735-8. MR 1036734.
Mas-Colell, Whinston & Green (1995, pp. 627–630): Mas-Colell, Andreu· Whinston, Michael D.· Green, Jerry R. (1995). «17.1 Large economies and nonconvexities». Microeconomic theory. Oxford University Press. ISBN 978-0-19-507340-9.
Arrow & Hahn (1980, pp. 169–182)
Mas-Colell (1985, pp. 52–55, 145–146, 152–153, and 274–275): Mas-Colell, Andreu (1985). «1.L Averages of sets». The Theory of general economic equilibrium: A differentiable approach. Econometric Society monographs. 9. Cambridge University Press. ISBN 0-521-26514-2. MR 1113262.
Hildenbrand (1974, pp. 37, 115–116, 122, and 168): Hildenbrand, Werner (1974). Core and equilibria of a large economy. Princeton studies in mathematical economics. 5. Princeton, N.J.: Princeton University Press. σελίδες viii+251. ISBN 978-0-691-04189-6. MR 0389160.
Arrow, Kenneth J.· Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 έκδοση). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057.
Starr (1997, p. 169): Starr, Ross M. (1997). «8 Convex sets, separation theorems, and non-convex sets in R^N (new chapters 22 and 25–26 in (2011) second ed.)». General equilibrium theory: An introduction (1η έκδοση). Cambridge: Cambridge University Press. σελίδες xxiii+250. ISBN 0-521-56473-5. MR 1462618.
Ellickson (1994, pp. xviii, 306–310, 312, 328–329, 347, and 352): Ellickson, Bryan (1994). Competitive equilibrium: Theory and applications. Cambridge University Press. doi:10.2277/0521319889. ISBN 978-0-521-31988-1.
Ichiishi (1983, pp. 24–25): Ichiishi, Tatsuro (1983). Game theory for economic analysis. Economic theory, econometrics, and mathematical economics. New York: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]. σελίδες x+164. ISBN 0-12-370180-5. MR 0700688.
Cassels (1981, pp. 127 and 33–34): Cassels, J. W. S. (1981). «Appendix A Convex sets». Economics for mathematicians. London Mathematical Society lecture note series. 62. Cambridge, New York: Cambridge University Press. σελίδες xi+145. ISBN 0-521-28614-X. MR 0657578.
Aubin (2007, pp. 458–476): Aubin, Jean-Pierre (2007). «14.2 Duality in the case of non-convex integral criterion and constraints (especially 14.2.3 The Shapley–Folkman theorem, pages 463–465)». Mathematical methods of game and economic theory (Reprint with new preface of 1982 North-Holland revised English έκδοση). Mineola, NY: Dover Publications, Inc. σελίδες xxxii+616. ISBN 978-0-486-46265-3. MR 2449499.
Carter (2001, pp. 93–94, 143, 318–319, 375–377, and 416) Carter, Michael (2001). Foundations of mathematical economics. Cambridge, MA: MIT Press. σελίδες xx+649. ISBN 0-262-53192-5. MR 1865841. (Author's website with answers to exercises). Αρχειοθετήθηκε από το πρωτότυπο στις 15 Σεπτεμβρίου 2006. Ανακτήθηκε στις 25 Ιουνίου 2014.
Trockel (1984, p. 30): Trockel, Walter (1984). Market demand: An analysis of large economies with nonconvex preferences. Lecture notes in economics and mathematical systems. 223. Berlin: Springer-Verlag. σελίδες viii+205. ISBN 3-540-12881-6. MR 0737006.
Rockafellar (1997, p. 23) Rockafellar, R. Tyrrell (1997). Convex analysis. Princeton Landmarks in Mathematics (Reprint of the 1970 (MR 274683) Princeton Mathematical Series 28 έκδοση). Princeton, NJ: Princeton University Press. σελίδες xviii+451. ISBN 0-691-01586-4. MR 1451876.
Lemaréchal (1973, p. 38): Lemaréchal, Claude (Απρίλιος 1973), Utilisation de la dualité dans les problémes non convexes [Use of duality for non–convex problems], Domaine de Voluceau, Rocquencourt, 78150 Le Chesnay, France: National Institute for Research in Computer Science and Control (IRIA), Laboratoire de recherche en informatique et automatique, σελ. 41 . Lemaréchal's experiments were discussed in later publications:
Aardal (1995, pp. 2–3): Aardal, Karen (Μάρτιος 1995). «Optima interview Claude Lemaréchal». Optima: Mathematical Programming Society newsletter 45: 2–4. http://www.mathprog.org/Old-Optima-Issues/optima45.pdf. Ανακτήθηκε στις 2011-02-02.
Hiriart-Urruty & Lemaréchal (1993, pp. 143–145, 151, 153, and 156): Hiriart-Urruty, Jean-Baptiste· Lemaréchal, Claude (1993). «XII Abstract duality for practitioners». Convex analysis and minimization algorithms, Volume II: Advanced theory and bundle methods. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. 306. Berlin: Springer-Verlag. σελίδες 136–193 (and bibliographical comments on pp.334–335). ISBN 3-540-56852-2. MR 1295240.
Ekeland, Ivar (1974). «Une estimation a priori en programmation non convexe» (στα γαλλικά). Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries A et B 279: 149–151. ISSN 0151-0509. MR 395844.
Aubin & Ekeland (1976, pp. 226, 233, 235, 238, and 241): Aubin, J. P.; Ekeland, I. (1976). «Estimates of the duality gap in nonconvex optimization». Mathematics of Operations Research 1 (3): 225–245. doi:10.1287/moor.1.3.225. MR 449695.
Aubin & Ekeland (1976) and Ekeland (1999, pp. 362–364) also considered the convex closure of a problem of non-convex minimization—that is, the problem defined as the closed convex hull of the epigraph of the original problem. Their study of duality gaps was extended by Di Guglielmo to the quasiconvex closure of a non-convex minimization problem—that is, the problem defined as the closed convex hull of the lower level sets:
Di Guglielmo (1977, pp. 287–288): Di Guglielmo, F. (1977). «Nonconvex duality in multiobjective optimization». Mathematics of Operations Research 2 (3): 285–291. doi:10.1287/moor.2.3.285. MR 484418.

Ekeland, Ivar (1999) [1976]. «Appendix I: An a priori estimate in convex programming». Στο: Ekeland, Ivar· Temam, Roger. Convex analysis and variational problems. Classics in Applied Mathematics. 28 (Corrected reprinting of the North-Holland έκδοση). Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). σελίδες 357–373. ISBN 0-89871-450-8. MR 1727362.
Schneider & Weil (2008, p. 45): Schneider, Rolf· Weil, Wolfgang (2008). Stochastic and integral geometry. Probability and its applications. Springer. doi:10.1007/978-3-540-78859-1. ISBN 978-3-540-78858-4. MR 2455326. Αρχειοθετήθηκε από το πρωτότυπο στις 9 Οκτωβρίου 2012. Ανακτήθηκε στις 25 Ιουνίου 2014.
Cassels (1975, pp. 433–434): Cassels, J. W. S. (1975). «Measures of the non-convexity of sets and the Shapley–Folkman–Starr theorem». Mathematical Proceedings of the Cambridge Philosophical Society 78 (3): 433–436. doi:10.1017/S0305004100051884. MR 385711. http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2075868&fulltextType=RA&fileId=S0305004100051884ER.
Molchanov (2005, pp. 195–198, 218, 232, 237–238 and 407): Molchanov, Ilya (2005). «3 Minkowski addition». Theory of random sets. Probability and its applications. London: Springer-Verlag London Ltd. σελίδες 194–240. doi:10.1007/1-84628-150-4. ISBN 978-1-84996-949-9. MR 2132405. ^{[νεκρός σύνδεσμος]}
Puri & Ralescu (1985, pp. 154–155): Puri, Madan L.; Ralescu, Dan A. (1985). «Limit theorems for random compact sets in Banach space». Mathematical Proceedings of the Cambridge Philosophical Society 97 (1): 151–158. doi:10.1017/S0305004100062691. MR 764504. http://journals.cambridge.org/action/displayAbstract?aid=2087952.
Weil (1982, pp. 203, and 205–206): Weil, Wolfgang (1982). «An application of the central limit theorem for Banach-space–valued random variables to the theory of random sets». Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete [Probability Theory and Related Fields] 60 (2): 203–208. doi:10.1007/BF00531823. MR 663901.
Cerf (1999, pp. 243–244): Cerf, Raphaël (1999). «Large deviations for sums of i.i.d. random compact sets». Proceedings of the American Mathematical Society 127 (8): 2431–2436. doi:10.1090/S0002-9939-99-04788-7. MR 1487361. http://www.ams.org/journals/proc/1999-127-08/S0002-9939-99-04788-7. Cerf uses applications of the Shapley–Folkman lemma from Puri & Ralescu (1985, pp. 154–155).
Ruzsa (1997, p. 345): Ruzsa, Imre Z. (1997). «The Brunn–Minkowski inequality and nonconvex sets». Geometriae Dedicata 67 (3): 337–348. doi:10.1023/A:1004958110076. MR 1475877.
Tardella (1990, pp. 478–479): Tardella, Fabio (1990). «A new proof of the Lyapunov convexity theorem». SIAM Journal on Control and Optimization 28 (2): 478–481. doi:10.1137/0328026. MR 1040471.
Mas-Colell (1978, p. 210): Mas-Colell, Andreu (1978). «A note on the core equivalence theorem: How many blocking coalitions are there?». Journal of Mathematical Economics 5 (3): 207–215. doi:10.1016/0304-4068(78)90010-1. MR 514468. Αρχειοθετήθηκε από το πρωτότυπο στις 2013-02-02. https://archive.today/20130202023110/http://www.sciencedirect.com/science/article/pii/0304406878900101. Ανακτήθηκε στις 2014-06-25.

archive.org

Guesnerie (1989, p. 138) Guesnerie, Roger (1989). «First-best allocation of resources with nonconvexities in production». Στο: Cornet, Bernard· Tulkens, Henry. Contributions to Operations Research and Economics: The twentieth anniversary of CORE (Papers from the symposium held in Louvain-la-Neuve, January 1987). Cambridge, MA: MIT Press. σελίδες 99–143. ISBN 0-262-03149-3. MR 1104662.
Wold (1943b, pp. 231 and 239–240): Wold, Herman (1943b). «A synthesis of pure demand analysis II». Skandinavisk Aktuarietidskrift [Scandinavian Actuarial Journal] 26: 220–263. MR 11939.
Wold & Juréen (1953, p. 146): Wold, Herman· Juréen, Lars (in association with Wold) (1953). «8 Some further applications of preference fields (pp. 129–148)». Demand analysis: A study in econometrics. Wiley publications in statistics. New York: John Wiley and Sons, Inc. σελίδες xvi+358. MR 0064385.
Farrell, M. J. (Αύγουστος 1959). «The Convexity assumption in the theory of competitive markets». The Journal of Political Economy 67 (4): 371–391. doi:10.1086/258197. https://archive.org/details/sim_journal-of-political-economy_1959-08_67_4/page/371. Farrell, M. J. (Οκτώβριος 1961a). On Convexity, efficiency, and markets: A Reply. 69, σελ. 484–489. Farrell, M. J. (Οκτώβριος 1961b). The Convexity assumption in the theory of competitive markets: Rejoinder. 69, σελ. 493.
Koopmans, Tjalling C. (Οκτώβριος 1961). «Convexity assumptions, allocative efficiency, and competitive equilibrium». The Journal of Political Economy 69 (5): 478–479. doi:10.1086/258539. https://archive.org/details/sim_journal-of-political-economy_1961-10_69_5/page/478.
Koopmans (1961, p. 478) and others—for example, Farrell (1959, pp. 390–391) and Farrell (1961a, p. 484), Bator (1961, pp. 482–483), Rothenberg (1960, p. 438), and Starr (1969, p. 26)—commented on Koopmans (1957, pp. 1–126, especially 9–16 [1.3 Summation of opportunity sets], 23–35 [1.6 Convex sets and the price implications of optimality], and 35–37 [1.7 The role of convexity assumptions in the analysis]):
Tjalling C., Koopmans (1957). «Allocation of resources and the price system». Στο: Koopmans, Tjalling C. Three essays on the state of economic science. New York: McGraw–Hill Book Company. σελίδες 1–126. ISBN 0-07-035337-9.
Rothenberg (1960, p. 447): Rothenberg, Jerome (Οκτώβριος 1960). «Non-convexity, aggregation, and Pareto optimality». The Journal of Political Economy 68 (5): 435–468. doi:10.1086/258363. https://archive.org/details/sim_journal-of-political-economy_1960-10_68_5/page/435. (Rothenberg, Jerome (Οκτώβριος 1961). Comments on non-convexity. 69, σελ. 490–492. )
Shapley & Shubik (1966, p. 806): Shapley, L. S.; Shubik, M. (Οκτώβριος 1966). «Quasi-cores in a monetary economy with nonconvex preferences». Econometrica 34 (4): 805–827. doi:10.2307/1910101. Zbl 154.45303. https://archive.org/details/sim_econometrica_1966-10_34_4/page/805.
Aumann (1966, pp. 1–2): Aumann, Robert J. (January 1966). «Existence of competitive equilibrium in markets with a continuum of traders». Econometrica 34 (1): 1–17. MR 191623. https://archive.org/details/sim_econometrica_1966-01_34_1/page/1. Aumann (1966) uses results from Aumann (1964, 1965): Aumann, Robert J. (Ιανουάριος–Απρίλιος 1964). «Markets with a continuum of traders». Econometrica 32 (1–2): 39–50. MR 172689. Aumann, Robert J. (Αύγουστος 1965). «Integrals of set-valued functions». Journal of Mathematical Analysis and Applications 12 (1): 1–12. doi:10.1016/0022-247X(65)90049-1. MR 185073. Αρχειοθετήθηκε από το πρωτότυπο στις 2013-02-02. https://archive.today/20130202032422/http://www.sciencedirect.com/science/article/pii/0022247X65900491. Ανακτήθηκε στις 2014-06-25.
Guesnerie (1989, pp. 99) Guesnerie, Roger (1989). «First-best allocation of resources with nonconvexities in production». Στο: Cornet, Bernard· Tulkens, Henry. Contributions to Operations Research and Economics: The twentieth anniversary of CORE (Papers from the symposium held in Louvain-la-Neuve, January 1987). Cambridge, MA: MIT Press. σελίδες 99–143. ISBN 0-262-03149-3. MR 1104662.
Varian (1992, pp. 393–394): Varian, Hal R. (1992). «21.2 Convexity and size». Microeconomic Analysis (3η έκδοση). W. W. Norton & Company. ISBN 978-0-393-95735-8. MR 1036734.
Mas-Colell, Whinston & Green (1995, pp. 627–630): Mas-Colell, Andreu· Whinston, Michael D.· Green, Jerry R. (1995). «17.1 Large economies and nonconvexities». Microeconomic theory. Oxford University Press. ISBN 978-0-19-507340-9.
Starr (1997, p. 169): Starr, Ross M. (1997). «8 Convex sets, separation theorems, and non-convex sets in R^N (new chapters 22 and 25–26 in (2011) second ed.)». General equilibrium theory: An introduction (1η έκδοση). Cambridge: Cambridge University Press. σελίδες xxiii+250. ISBN 0-521-56473-5. MR 1462618.
Ellickson (1994, pp. xviii, 306–310, 312, 328–329, 347, and 352): Ellickson, Bryan (1994). Competitive equilibrium: Theory and applications. Cambridge University Press. doi:10.2277/0521319889. ISBN 978-0-521-31988-1.
Laffont (1988, pp. 63–65): Laffont, Jean-Jacques (1988). «3 Nonconvexities». Fundamentals of public economics. MIT. ISBN 0-262-12127-1.
Salanié (2000, pp. 112–113 and 107–115): Salanié, Bernard (2000). «7 Nonconvexities». Microeconomics of market failures (English translation of the (1998) French Microéconomie: Les défaillances du marché (Economica, Paris) έκδοση). Cambridge, MA: MIT Press. σελίδες 107–125. ISBN 0-262-19443-0.
Ichiishi (1983, pp. 24–25): Ichiishi, Tatsuro (1983). Game theory for economic analysis. Economic theory, econometrics, and mathematical economics. New York: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]. σελίδες x+164. ISBN 0-12-370180-5. MR 0700688.
Vind (1964, pp. 168 and 175): Vind, Karl (Μάιος 1964). «Edgeworth-allocations in an exchange economy with many traders». International Economic Review 5 (2): 165–77. https://archive.org/details/sim_international-economic-review_1964-05_5_2/page/165. Vind's article was noted by the winner of the 1983 Nobel Prize in Economics, Gérard Debreu Debreu (1991, p. 4) wrote:

The concept of a convex set (i.e., a set containing the segment connecting any two of its points) had repeatedly been placed at the center of economic theory before 1964. It appeared in a new light with the introduction of integration theory in the study of economic competition: If one associates with every agent of an economy an arbitrary set in the commodity space and if one averages those individual sets over a collection of insignificant agents, then the resulting set is necessarily convex. [Debreu appends this footnote: "On this direct consequence of a theorem of A. A. Lyapunov, see Vind (1964)."] But explanations of the ... functions of prices ... can be made to rest on the convexity of sets derived by that averaging process. Convexity in the commodity space obtained by aggregation over a collection of insignificant agents is an insight that economic theory owes ... to integration theory. [Italics added]

Debreu, Gérard (Μάρτιος 1991). «The Mathematization of economic theory». The American Economic Review 81 (Presidential address delivered at the 103rd meeting of the American Economic Association, 29 December 1990, Washington, DC): 1–7.

archive.today

Arrow & Hahn (1980, p. 376), Rockafellar (1997, pp. 10–11), and Green & Heller (1981, p. 37) Arrow, Kenneth J.· Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 έκδοση). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057. Rockafellar, R. Tyrrell (1997). Convex analysis. Princeton Landmarks in Mathematics (Reprint of the 1970 (MR 274683) Princeton Mathematical Series 28 έκδοση). Princeton, NJ: Princeton University Press. σελίδες xviii+451. ISBN 0-691-01586-4. MR 1451876. Green, Jerry· Heller, Walter P. (1981). «1 Mathematical analysis and convexity with applications to economics». Στο: Arrow, Kenneth Joseph· Intriligator, Michael D. Handbook of mathematical economics, Volume I. Handbooks in Economics. 1. Amsterdam: North-Holland Publishing Co. σελίδες 15–52. doi:10.1016/S1573-4382(81)01005-9. ISBN 0-444-86126-2. MR 0634800. Αρχειοθετήθηκε από το πρωτότυπο στις 17 Δεκεμβρίου 2012. Ανακτήθηκε στις 25 Ιουνίου 2014.
Starr, Ross M. (1981). «Approximation of points of convex hull of a sum of sets by points of the sum: An elementary approach». Journal of Economic Theory 25 (2): 314–317. doi:10.1016/0022-0531(81)90010-7. MR 640201. Αρχειοθετήθηκε από το πρωτότυπο στις 2013-02-02. https://archive.today/20130202094407/http://www.sciencedirect.com/science/article/pii/0022053181900107. Ανακτήθηκε στις 2014-06-25.
Diewert (1982, pp. 552–553) Diewert, W. E. (1982). «12 Duality approaches to microeconomic theory». Στο: Arrow, Kenneth Joseph· Intriligator, Michael D. Handbook of mathematical economics, Volume II. Handbooks in Economics. 1. Amsterdam: North-Holland Publishing Co. σελίδες 535–599. doi:10.1016/S1573-4382(82)02007-4. ISBN 978-0-444-86127-6. MR 0648778. Αρχειοθετήθηκε από το πρωτότυπο στις 10 Δεκεμβρίου 2012. Ανακτήθηκε στις 25 Ιουνίου 2014.
Aumann (1966, pp. 1–2): Aumann, Robert J. (January 1966). «Existence of competitive equilibrium in markets with a continuum of traders». Econometrica 34 (1): 1–17. MR 191623. https://archive.org/details/sim_econometrica_1966-01_34_1/page/1. Aumann (1966) uses results from Aumann (1964, 1965): Aumann, Robert J. (Ιανουάριος–Απρίλιος 1964). «Markets with a continuum of traders». Econometrica 32 (1–2): 39–50. MR 172689. Aumann, Robert J. (Αύγουστος 1965). «Integrals of set-valued functions». Journal of Mathematical Analysis and Applications 12 (1): 1–12. doi:10.1016/0022-247X(65)90049-1. MR 185073. Αρχειοθετήθηκε από το πρωτότυπο στις 2013-02-02. https://archive.today/20130202032422/http://www.sciencedirect.com/science/article/pii/0022247X65900491. Ανακτήθηκε στις 2014-06-25.
Taking the convex hull of non-convex preferences had been discussed earlier by Wold (1943b, p. 243) and by Wold & Juréen (1953, p. 146), according to Diewert (1982, p. 552). Diewert, W. E. (1982). «12 Duality approaches to microeconomic theory». Στο: Arrow, Kenneth Joseph· Intriligator, Michael D. Handbook of mathematical economics, Volume II. Handbooks in Economics. 1. Amsterdam: North-Holland Publishing Co. σελίδες 535–599. doi:10.1016/S1573-4382(82)02007-4. ISBN 978-0-444-86127-6. MR 0648778. Αρχειοθετήθηκε από το πρωτότυπο στις 10 Δεκεμβρίου 2012. Ανακτήθηκε στις 25 Ιουνίου 2014.
Green & Heller (1981, p. 44) Green, Jerry· Heller, Walter P. (1981). «1 Mathematical analysis and convexity with applications to economics». Στο: Arrow, Kenneth Joseph· Intriligator, Michael D. Handbook of mathematical economics, Volume I. Handbooks in Economics. 1. Amsterdam: North-Holland Publishing Co. σελίδες 15–52. doi:10.1016/S1573-4382(81)01005-9. ISBN 0-444-86126-2. MR 0634800. Αρχειοθετήθηκε από το πρωτότυπο στις 17 Δεκεμβρίου 2012. Ανακτήθηκε στις 25 Ιουνίου 2014.
Mas-Colell (1978, p. 210): Mas-Colell, Andreu (1978). «A note on the core equivalence theorem: How many blocking coalitions are there?». Journal of Mathematical Economics 5 (3): 207–215. doi:10.1016/0304-4068(78)90010-1. MR 514468. Αρχειοθετήθηκε από το πρωτότυπο στις 2013-02-02. https://archive.today/20130202023110/http://www.sciencedirect.com/science/article/pii/0304406878900101. Ανακτήθηκε στις 2014-06-25.

berkeley.edu

elsa.berkeley.edu

Anderson, Robert M. (2005-03-14), «1 The Shapley–Folkman theorem», Economics 201B: Nonconvex preferences and approximate equilibria, Berkeley, CA: Economics Department, University of California, Berkeley, σελ. 1–5, http://elsa.berkeley.edu/users/anderson/Econ201B/NonconvexHandout.pdf, ανακτήθηκε στις 2011-01-01

cambridge.org

journals.cambridge.org

Cassels (1975, pp. 433–434): Cassels, J. W. S. (1975). «Measures of the non-convexity of sets and the Shapley–Folkman–Starr theorem». Mathematical Proceedings of the Cambridge Philosophical Society 78 (3): 433–436. doi:10.1017/S0305004100051884. MR 385711. http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2075868&fulltextType=RA&fileId=S0305004100051884ER.
Puri & Ralescu (1985, pp. 154–155): Puri, Madan L.; Ralescu, Dan A. (1985). «Limit theorems for random compact sets in Banach space». Mathematical Proceedings of the Cambridge Philosophical Society 97 (1): 151–158. doi:10.1017/S0305004100062691. MR 764504. http://journals.cambridge.org/action/displayAbstract?aid=2087952.

dictionaryofeconomics.com

Starr (2008) Starr, Ross M. (2008). «Shapley–Folkman theorem». Στο: Durlauf, Steven N.· Blume, Lawrence E. The new Palgrave dictionary of economics (1η έκδοση). Palgrave Macmillan. σελίδες 317–318. doi:10.1057/9780230226203.1518.
Mas-Colell (1987) Mas-Colell, A. (1987). «Non-convexity». Στο: Eatwell, John· Milgate, Murray· Newman, Peter. The new Palgrave: A dictionary of economics (1η έκδοση). Palgrave Macmillan. σελίδες 653–661. doi:10.1057/9780230226203.3173. (PDF file at Mas-Colell's homepage).

doi.org

dx.doi.org

Artstein & Vitale (1975, pp. 881–882): Artstein, Zvi; Vitale, Richard A. (1975), «A strong law of large numbers for random compact sets», The Annals of Probability 3 (5): 879–882, doi:10.1214/aop/1176996275, Πρότυπο:Euclid, Zbl 0313.60012, http://projecteuclid.org/euclid.aop/1176996275
Starr, Ross M. (1981). «Approximation of points of convex hull of a sum of sets by points of the sum: An elementary approach». Journal of Economic Theory 25 (2): 314–317. doi:10.1016/0022-0531(81)90010-7. MR 640201. Αρχειοθετήθηκε από το πρωτότυπο στις 2013-02-02. https://archive.today/20130202094407/http://www.sciencedirect.com/science/article/pii/0022053181900107. Ανακτήθηκε στις 2014-06-25.
Farrell, M. J. (Αύγουστος 1959). «The Convexity assumption in the theory of competitive markets». The Journal of Political Economy 67 (4): 371–391. doi:10.1086/258197. https://archive.org/details/sim_journal-of-political-economy_1959-08_67_4/page/371. Farrell, M. J. (Οκτώβριος 1961a). On Convexity, efficiency, and markets: A Reply. 69, σελ. 484–489. Farrell, M. J. (Οκτώβριος 1961b). The Convexity assumption in the theory of competitive markets: Rejoinder. 69, σελ. 493.
Bator, Francis M. (Οκτώβριος 1961a). «On convexity, efficiency, and markets». The Journal of Political Economy 69 (5): 480–483. doi:10.1086/258540. Bator, Francis M. (Οκτώβριος 1961b). On convexity, efficiency, and markets: Rejoinder. 69, σελ. 489.
Koopmans, Tjalling C. (Οκτώβριος 1961). «Convexity assumptions, allocative efficiency, and competitive equilibrium». The Journal of Political Economy 69 (5): 478–479. doi:10.1086/258539. https://archive.org/details/sim_journal-of-political-economy_1961-10_69_5/page/478.
Koopmans (1961, p. 478) and others—for example, Farrell (1959, pp. 390–391) and Farrell (1961a, p. 484), Bator (1961, pp. 482–483), Rothenberg (1960, p. 438), and Starr (1969, p. 26)—commented on Koopmans (1957, pp. 1–126, especially 9–16 [1.3 Summation of opportunity sets], 23–35 [1.6 Convex sets and the price implications of optimality], and 35–37 [1.7 The role of convexity assumptions in the analysis]):
Tjalling C., Koopmans (1957). «Allocation of resources and the price system». Στο: Koopmans, Tjalling C. Three essays on the state of economic science. New York: McGraw–Hill Book Company. σελίδες 1–126. ISBN 0-07-035337-9.
Rothenberg (1960, p. 447): Rothenberg, Jerome (Οκτώβριος 1960). «Non-convexity, aggregation, and Pareto optimality». The Journal of Political Economy 68 (5): 435–468. doi:10.1086/258363. https://archive.org/details/sim_journal-of-political-economy_1960-10_68_5/page/435. (Rothenberg, Jerome (Οκτώβριος 1961). Comments on non-convexity. 69, σελ. 490–492. )
Shapley & Shubik (1966, p. 806): Shapley, L. S.; Shubik, M. (Οκτώβριος 1966). «Quasi-cores in a monetary economy with nonconvex preferences». Econometrica 34 (4): 805–827. doi:10.2307/1910101. Zbl 154.45303. https://archive.org/details/sim_econometrica_1966-10_34_4/page/805.
Aumann (1966, pp. 1–2): Aumann, Robert J. (January 1966). «Existence of competitive equilibrium in markets with a continuum of traders». Econometrica 34 (1): 1–17. MR 191623. https://archive.org/details/sim_econometrica_1966-01_34_1/page/1. Aumann (1966) uses results from Aumann (1964, 1965): Aumann, Robert J. (Ιανουάριος–Απρίλιος 1964). «Markets with a continuum of traders». Econometrica 32 (1–2): 39–50. MR 172689. Aumann, Robert J. (Αύγουστος 1965). «Integrals of set-valued functions». Journal of Mathematical Analysis and Applications 12 (1): 1–12. doi:10.1016/0022-247X(65)90049-1. MR 185073. Αρχειοθετήθηκε από το πρωτότυπο στις 2013-02-02. https://archive.today/20130202032422/http://www.sciencedirect.com/science/article/pii/0022247X65900491. Ανακτήθηκε στις 2014-06-25.
Aubin & Ekeland (1976, pp. 226, 233, 235, 238, and 241): Aubin, J. P.; Ekeland, I. (1976). «Estimates of the duality gap in nonconvex optimization». Mathematics of Operations Research 1 (3): 225–245. doi:10.1287/moor.1.3.225. MR 449695.
Aubin & Ekeland (1976) and Ekeland (1999, pp. 362–364) also considered the convex closure of a problem of non-convex minimization—that is, the problem defined as the closed convex hull of the epigraph of the original problem. Their study of duality gaps was extended by Di Guglielmo to the quasiconvex closure of a non-convex minimization problem—that is, the problem defined as the closed convex hull of the lower level sets:
Di Guglielmo (1977, pp. 287–288): Di Guglielmo, F. (1977). «Nonconvex duality in multiobjective optimization». Mathematics of Operations Research 2 (3): 285–291. doi:10.1287/moor.2.3.285. MR 484418.

Ekeland, Ivar (1999) [1976]. «Appendix I: An a priori estimate in convex programming». Στο: Ekeland, Ivar· Temam, Roger. Convex analysis and variational problems. Classics in Applied Mathematics. 28 (Corrected reprinting of the North-Holland έκδοση). Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). σελίδες 357–373. ISBN 0-89871-450-8. MR 1727362.
Cassels (1975, pp. 433–434): Cassels, J. W. S. (1975). «Measures of the non-convexity of sets and the Shapley–Folkman–Starr theorem». Mathematical Proceedings of the Cambridge Philosophical Society 78 (3): 433–436. doi:10.1017/S0305004100051884. MR 385711. http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2075868&fulltextType=RA&fileId=S0305004100051884ER.
Puri & Ralescu (1985, pp. 154–155): Puri, Madan L.; Ralescu, Dan A. (1985). «Limit theorems for random compact sets in Banach space». Mathematical Proceedings of the Cambridge Philosophical Society 97 (1): 151–158. doi:10.1017/S0305004100062691. MR 764504. http://journals.cambridge.org/action/displayAbstract?aid=2087952.
Weil (1982, pp. 203, and 205–206): Weil, Wolfgang (1982). «An application of the central limit theorem for Banach-space–valued random variables to the theory of random sets». Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete [Probability Theory and Related Fields] 60 (2): 203–208. doi:10.1007/BF00531823. MR 663901.
Cerf (1999, pp. 243–244): Cerf, Raphaël (1999). «Large deviations for sums of i.i.d. random compact sets». Proceedings of the American Mathematical Society 127 (8): 2431–2436. doi:10.1090/S0002-9939-99-04788-7. MR 1487361. http://www.ams.org/journals/proc/1999-127-08/S0002-9939-99-04788-7. Cerf uses applications of the Shapley–Folkman lemma from Puri & Ralescu (1985, pp. 154–155).
Ruzsa (1997, p. 345): Ruzsa, Imre Z. (1997). «The Brunn–Minkowski inequality and nonconvex sets». Geometriae Dedicata 67 (3): 337–348. doi:10.1023/A:1004958110076. MR 1475877.
Tardella (1990, pp. 478–479): Tardella, Fabio (1990). «A new proof of the Lyapunov convexity theorem». SIAM Journal on Control and Optimization 28 (2): 478–481. doi:10.1137/0328026. MR 1040471.
Mas-Colell (1978, p. 210): Mas-Colell, Andreu (1978). «A note on the core equivalence theorem: How many blocking coalitions are there?». Journal of Mathematical Economics 5 (3): 207–215. doi:10.1016/0304-4068(78)90010-1. MR 514468. Αρχειοθετήθηκε από το πρωτότυπο στις 2013-02-02. https://archive.today/20130202023110/http://www.sciencedirect.com/science/article/pii/0304406878900101. Ανακτήθηκε στις 2014-06-25.

doi.org

Starr (2008) Starr, Ross M. (2008). «Shapley–Folkman theorem». Στο: Durlauf, Steven N.· Blume, Lawrence E. The new Palgrave dictionary of economics (1η έκδοση). Palgrave Macmillan. σελίδες 317–318. doi:10.1057/9780230226203.1518.
Arrow & Hahn (1980, p. 376), Rockafellar (1997, pp. 10–11), and Green & Heller (1981, p. 37) Arrow, Kenneth J.· Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 έκδοση). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057. Rockafellar, R. Tyrrell (1997). Convex analysis. Princeton Landmarks in Mathematics (Reprint of the 1970 (MR 274683) Princeton Mathematical Series 28 έκδοση). Princeton, NJ: Princeton University Press. σελίδες xviii+451. ISBN 0-691-01586-4. MR 1451876. Green, Jerry· Heller, Walter P. (1981). «1 Mathematical analysis and convexity with applications to economics». Στο: Arrow, Kenneth Joseph· Intriligator, Michael D. Handbook of mathematical economics, Volume I. Handbooks in Economics. 1. Amsterdam: North-Holland Publishing Co. σελίδες 15–52. doi:10.1016/S1573-4382(81)01005-9. ISBN 0-444-86126-2. MR 0634800. Αρχειοθετήθηκε από το πρωτότυπο στις 17 Δεκεμβρίου 2012. Ανακτήθηκε στις 25 Ιουνίου 2014.
Diewert (1982, pp. 552–553) Diewert, W. E. (1982). «12 Duality approaches to microeconomic theory». Στο: Arrow, Kenneth Joseph· Intriligator, Michael D. Handbook of mathematical economics, Volume II. Handbooks in Economics. 1. Amsterdam: North-Holland Publishing Co. σελίδες 535–599. doi:10.1016/S1573-4382(82)02007-4. ISBN 978-0-444-86127-6. MR 0648778. Αρχειοθετήθηκε από το πρωτότυπο στις 10 Δεκεμβρίου 2012. Ανακτήθηκε στις 25 Ιουνίου 2014.
Taking the convex hull of non-convex preferences had been discussed earlier by Wold (1943b, p. 243) and by Wold & Juréen (1953, p. 146), according to Diewert (1982, p. 552). Diewert, W. E. (1982). «12 Duality approaches to microeconomic theory». Στο: Arrow, Kenneth Joseph· Intriligator, Michael D. Handbook of mathematical economics, Volume II. Handbooks in Economics. 1. Amsterdam: North-Holland Publishing Co. σελίδες 535–599. doi:10.1016/S1573-4382(82)02007-4. ISBN 978-0-444-86127-6. MR 0648778. Αρχειοθετήθηκε από το πρωτότυπο στις 10 Δεκεμβρίου 2012. Ανακτήθηκε στις 25 Ιουνίου 2014.
Starr & Stinchcombe (1999, pp. 217–218): Starr, R. M.· Stinchcombe, M. B. (1999). «Exchange in a network of trading posts». Στο: Chichilnisky, Graciela. Markets, information and uncertainty: Essays in economic theory in honor of Kenneth J. Arrow. Cambridge: Cambridge University Press. σελίδες 217–234. doi:10.2277/0521553555. ISBN 978-0-521-08288-4.
Green & Heller (1981, p. 44) Green, Jerry· Heller, Walter P. (1981). «1 Mathematical analysis and convexity with applications to economics». Στο: Arrow, Kenneth Joseph· Intriligator, Michael D. Handbook of mathematical economics, Volume I. Handbooks in Economics. 1. Amsterdam: North-Holland Publishing Co. σελίδες 15–52. doi:10.1016/S1573-4382(81)01005-9. ISBN 0-444-86126-2. MR 0634800. Αρχειοθετήθηκε από το πρωτότυπο στις 17 Δεκεμβρίου 2012. Ανακτήθηκε στις 25 Ιουνίου 2014.
Mas-Colell (1987) Mas-Colell, A. (1987). «Non-convexity». Στο: Eatwell, John· Milgate, Murray· Newman, Peter. The new Palgrave: A dictionary of economics (1η έκδοση). Palgrave Macmillan. σελίδες 653–661. doi:10.1057/9780230226203.3173. (PDF file at Mas-Colell's homepage).
Starr (1997, p. 169): Starr, Ross M. (1997). «8 Convex sets, separation theorems, and non-convex sets in R^N (new chapters 22 and 25–26 in (2011) second ed.)». General equilibrium theory: An introduction (1η έκδοση). Cambridge: Cambridge University Press. σελίδες xxiii+250. ISBN 0-521-56473-5. MR 1462618.
Ellickson (1994, pp. xviii, 306–310, 312, 328–329, 347, and 352): Ellickson, Bryan (1994). Competitive equilibrium: Theory and applications. Cambridge University Press. doi:10.2277/0521319889. ISBN 978-0-521-31988-1.
Schneider & Weil (2008, p. 45): Schneider, Rolf· Weil, Wolfgang (2008). Stochastic and integral geometry. Probability and its applications. Springer. doi:10.1007/978-3-540-78859-1. ISBN 978-3-540-78858-4. MR 2455326. Αρχειοθετήθηκε από το πρωτότυπο στις 9 Οκτωβρίου 2012. Ανακτήθηκε στις 25 Ιουνίου 2014.
Molchanov (2005, pp. 195–198, 218, 232, 237–238 and 407): Molchanov, Ilya (2005). «3 Minkowski addition». Theory of random sets. Probability and its applications. London: Springer-Verlag London Ltd. σελίδες 194–240. doi:10.1007/1-84628-150-4. ISBN 978-1-84996-949-9. MR 2132405. ^{[νεκρός σύνδεσμος]}

mathprog.org

Lemaréchal (1973, p. 38): Lemaréchal, Claude (Απρίλιος 1973), Utilisation de la dualité dans les problémes non convexes [Use of duality for non–convex problems], Domaine de Voluceau, Rocquencourt, 78150 Le Chesnay, France: National Institute for Research in Computer Science and Control (IRIA), Laboratoire de recherche en informatique et automatique, σελ. 41 . Lemaréchal's experiments were discussed in later publications:
Aardal (1995, pp. 2–3): Aardal, Karen (Μάρτιος 1995). «Optima interview Claude Lemaréchal». Optima: Mathematical Programming Society newsletter 45: 2–4. http://www.mathprog.org/Old-Optima-Issues/optima45.pdf. Ανακτήθηκε στις 2011-02-02.
Hiriart-Urruty & Lemaréchal (1993, pp. 143–145, 151, 153, and 156): Hiriart-Urruty, Jean-Baptiste· Lemaréchal, Claude (1993). «XII Abstract duality for practitioners». Convex analysis and minimization algorithms, Volume II: Advanced theory and bundle methods. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. 306. Berlin: Springer-Verlag. σελίδες 136–193 (and bibliographical comments on pp.334–335). ISBN 3-540-56852-2. MR 1295240.

michaelcarteronline.com

Carter (2001, p. 94) Carter, Michael (2001). Foundations of mathematical economics. Cambridge, MA: MIT Press. σελίδες xx+649. ISBN 0-262-53192-5. MR 1865841. (Author's website with answers to exercises). Αρχειοθετήθηκε από το πρωτότυπο στις 15 Σεπτεμβρίου 2006. Ανακτήθηκε στις 25 Ιουνίου 2014.
Carter (2001, pp. 93–94, 143, 318–319, 375–377, and 416) Carter, Michael (2001). Foundations of mathematical economics. Cambridge, MA: MIT Press. σελίδες xx+649. ISBN 0-262-53192-5. MR 1865841. (Author's website with answers to exercises). Αρχειοθετήθηκε από το πρωτότυπο στις 15 Σεπτεμβρίου 2006. Ανακτήθηκε στις 25 Ιουνίου 2014.

mit.edu

mitpress.mit.edu

Carter (2001, p. 94) Carter, Michael (2001). Foundations of mathematical economics. Cambridge, MA: MIT Press. σελίδες xx+649. ISBN 0-262-53192-5. MR 1865841. (Author's website with answers to exercises). Αρχειοθετήθηκε από το πρωτότυπο στις 15 Σεπτεμβρίου 2006. Ανακτήθηκε στις 25 Ιουνίου 2014.
Carter (2001, pp. 93–94, 143, 318–319, 375–377, and 416) Carter, Michael (2001). Foundations of mathematical economics. Cambridge, MA: MIT Press. σελίδες xx+649. ISBN 0-262-53192-5. MR 1865841. (Author's website with answers to exercises). Αρχειοθετήθηκε από το πρωτότυπο στις 15 Σεπτεμβρίου 2006. Ανακτήθηκε στις 25 Ιουνίου 2014.

web.mit.edu

Bertsekas (1996, pp. 364–381) acknowledging Ekeland (1999) on page 374 and Aubin & Ekeland (1976) on page 381:
Bertsekas, Dimitri P. (1996). «5.6 Large scale separable integer programming problems and the exponential method of multipliers». Constrained optimization and Lagrange multiplier methods (Reprint of (1982) Academic Press έκδοση). Belmont, MA: Athena Scientific. σελίδες xiii+395. ISBN 1-886529-04-3. MR 0690767.

Bertsekas (1996, pp. 364–381) describes an application of Lagrangian dual methods to the scheduling of electrical power plants ("unit commitment problems"), where non-convexity appears because of integer constraints:
Bertsekas, Dimitri P.; Lauer, Gregory S.; Sandell, Nils R., Jr.; Posbergh, Thomas A. (January 1983). «Optimal short-term scheduling of large-scale power systems». IEEE Transactions on Automatic Control AC-28 (Proceedings of 1981 IEEE Conference on Decision and Control, San Diego, CA, December 1981, pp.432–443): 1–11. http://web.mit.edu/dimitrib/www/Unit_Comm.pdf. Ανακτήθηκε στις 2011-02-02.
Ekeland, Ivar (1999) [1976]. «Appendix I: An a priori estimate in convex programming». Στο: Ekeland, Ivar· Temam, Roger. Convex analysis and variational problems. Classics in Applied Mathematics. 28 (Corrected reprinting of the North-Holland έκδοση). Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). σελίδες 357–373. ISBN 0-89871-450-8. MR 1727362.

projecteuclid.org

Artstein & Vitale (1975, pp. 881–882): Artstein, Zvi; Vitale, Richard A. (1975), «A strong law of large numbers for random compact sets», The Annals of Probability 3 (5): 879–882, doi:10.1214/aop/1176996275, Πρότυπο:Euclid, Zbl 0313.60012, http://projecteuclid.org/euclid.aop/1176996275

sciencedirect.com

Arrow & Hahn (1980, p. 376), Rockafellar (1997, pp. 10–11), and Green & Heller (1981, p. 37) Arrow, Kenneth J.· Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 έκδοση). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057. Rockafellar, R. Tyrrell (1997). Convex analysis. Princeton Landmarks in Mathematics (Reprint of the 1970 (MR 274683) Princeton Mathematical Series 28 έκδοση). Princeton, NJ: Princeton University Press. σελίδες xviii+451. ISBN 0-691-01586-4. MR 1451876. Green, Jerry· Heller, Walter P. (1981). «1 Mathematical analysis and convexity with applications to economics». Στο: Arrow, Kenneth Joseph· Intriligator, Michael D. Handbook of mathematical economics, Volume I. Handbooks in Economics. 1. Amsterdam: North-Holland Publishing Co. σελίδες 15–52. doi:10.1016/S1573-4382(81)01005-9. ISBN 0-444-86126-2. MR 0634800. Αρχειοθετήθηκε από το πρωτότυπο στις 17 Δεκεμβρίου 2012. Ανακτήθηκε στις 25 Ιουνίου 2014.
Starr, Ross M. (1981). «Approximation of points of convex hull of a sum of sets by points of the sum: An elementary approach». Journal of Economic Theory 25 (2): 314–317. doi:10.1016/0022-0531(81)90010-7. MR 640201. Αρχειοθετήθηκε από το πρωτότυπο στις 2013-02-02. https://archive.today/20130202094407/http://www.sciencedirect.com/science/article/pii/0022053181900107. Ανακτήθηκε στις 2014-06-25.
Diewert (1982, pp. 552–553) Diewert, W. E. (1982). «12 Duality approaches to microeconomic theory». Στο: Arrow, Kenneth Joseph· Intriligator, Michael D. Handbook of mathematical economics, Volume II. Handbooks in Economics. 1. Amsterdam: North-Holland Publishing Co. σελίδες 535–599. doi:10.1016/S1573-4382(82)02007-4. ISBN 978-0-444-86127-6. MR 0648778. Αρχειοθετήθηκε από το πρωτότυπο στις 10 Δεκεμβρίου 2012. Ανακτήθηκε στις 25 Ιουνίου 2014.
Aumann (1966, pp. 1–2): Aumann, Robert J. (January 1966). «Existence of competitive equilibrium in markets with a continuum of traders». Econometrica 34 (1): 1–17. MR 191623. https://archive.org/details/sim_econometrica_1966-01_34_1/page/1. Aumann (1966) uses results from Aumann (1964, 1965): Aumann, Robert J. (Ιανουάριος–Απρίλιος 1964). «Markets with a continuum of traders». Econometrica 32 (1–2): 39–50. MR 172689. Aumann, Robert J. (Αύγουστος 1965). «Integrals of set-valued functions». Journal of Mathematical Analysis and Applications 12 (1): 1–12. doi:10.1016/0022-247X(65)90049-1. MR 185073. Αρχειοθετήθηκε από το πρωτότυπο στις 2013-02-02. https://archive.today/20130202032422/http://www.sciencedirect.com/science/article/pii/0022247X65900491. Ανακτήθηκε στις 2014-06-25.
Taking the convex hull of non-convex preferences had been discussed earlier by Wold (1943b, p. 243) and by Wold & Juréen (1953, p. 146), according to Diewert (1982, p. 552). Diewert, W. E. (1982). «12 Duality approaches to microeconomic theory». Στο: Arrow, Kenneth Joseph· Intriligator, Michael D. Handbook of mathematical economics, Volume II. Handbooks in Economics. 1. Amsterdam: North-Holland Publishing Co. σελίδες 535–599. doi:10.1016/S1573-4382(82)02007-4. ISBN 978-0-444-86127-6. MR 0648778. Αρχειοθετήθηκε από το πρωτότυπο στις 10 Δεκεμβρίου 2012. Ανακτήθηκε στις 25 Ιουνίου 2014.
Green & Heller (1981, p. 44) Green, Jerry· Heller, Walter P. (1981). «1 Mathematical analysis and convexity with applications to economics». Στο: Arrow, Kenneth Joseph· Intriligator, Michael D. Handbook of mathematical economics, Volume I. Handbooks in Economics. 1. Amsterdam: North-Holland Publishing Co. σελίδες 15–52. doi:10.1016/S1573-4382(81)01005-9. ISBN 0-444-86126-2. MR 0634800. Αρχειοθετήθηκε από το πρωτότυπο στις 17 Δεκεμβρίου 2012. Ανακτήθηκε στις 25 Ιουνίου 2014.
Mas-Colell (1978, p. 210): Mas-Colell, Andreu (1978). «A note on the core equivalence theorem: How many blocking coalitions are there?». Journal of Mathematical Economics 5 (3): 207–215. doi:10.1016/0304-4068(78)90010-1. MR 514468. Αρχειοθετήθηκε από το πρωτότυπο στις 2013-02-02. https://archive.today/20130202023110/http://www.sciencedirect.com/science/article/pii/0304406878900101. Ανακτήθηκε στις 2014-06-25.

springerlink.com

Schneider & Weil (2008, p. 45): Schneider, Rolf· Weil, Wolfgang (2008). Stochastic and integral geometry. Probability and its applications. Springer. doi:10.1007/978-3-540-78859-1. ISBN 978-3-540-78858-4. MR 2455326. Αρχειοθετήθηκε από το πρωτότυπο στις 9 Οκτωβρίου 2012. Ανακτήθηκε στις 25 Ιουνίου 2014.
Molchanov (2005, pp. 195–198, 218, 232, 237–238 and 407): Molchanov, Ilya (2005). «3 Minkowski addition». Theory of random sets. Probability and its applications. London: Springer-Verlag London Ltd. σελίδες 194–240. doi:10.1007/1-84628-150-4. ISBN 978-1-84996-949-9. MR 2132405. ^{[νεκρός σύνδεσμος]}

umich.edu

press.umich.edu

Artstein (1980, pp. 172–183) Artstein (1980) was republished in a festschrift for Robert J. Aumann, winner of the 2008 Nobel Prize in Economics: Artstein, Zvi (1995). «22 Discrete and continuous bang–bang and facial spaces or: Look for the extreme points». Στο: Hart, Sergiu· Neyman, Abraham. Game and economic theory: Selected contributions in honor of Robert J. Aumann. Ann Arbor, MI: University of Michigan Press. σελίδες 449–462. ISBN 0-472-10673-2. Αρχειοθετήθηκε από το πρωτότυπο στις 24 Μαΐου 2011. Ανακτήθηκε στις 25 Ιουνίου 2014.

upf.edu

econ.upf.edu

Mas-Colell (1987) Mas-Colell, A. (1987). «Non-convexity». Στο: Eatwell, John· Milgate, Murray· Newman, Peter. The new Palgrave: A dictionary of economics (1η έκδοση). Palgrave Macmillan. σελίδες 653–661. doi:10.1057/9780230226203.3173. (PDF file at Mas-Colell's homepage).

web.archive.org

Carter (2001, p. 94) Carter, Michael (2001). Foundations of mathematical economics. Cambridge, MA: MIT Press. σελίδες xx+649. ISBN 0-262-53192-5. MR 1865841. (Author's website with answers to exercises). Αρχειοθετήθηκε από το πρωτότυπο στις 15 Σεπτεμβρίου 2006. Ανακτήθηκε στις 25 Ιουνίου 2014.
Carter (2001, pp. 93–94, 143, 318–319, 375–377, and 416) Carter, Michael (2001). Foundations of mathematical economics. Cambridge, MA: MIT Press. σελίδες xx+649. ISBN 0-262-53192-5. MR 1865841. (Author's website with answers to exercises). Αρχειοθετήθηκε από το πρωτότυπο στις 15 Σεπτεμβρίου 2006. Ανακτήθηκε στις 25 Ιουνίου 2014.
Schneider & Weil (2008, p. 45): Schneider, Rolf· Weil, Wolfgang (2008). Stochastic and integral geometry. Probability and its applications. Springer. doi:10.1007/978-3-540-78859-1. ISBN 978-3-540-78858-4. MR 2455326. Αρχειοθετήθηκε από το πρωτότυπο στις 9 Οκτωβρίου 2012. Ανακτήθηκε στις 25 Ιουνίου 2014.
Artstein (1980, pp. 172–183) Artstein (1980) was republished in a festschrift for Robert J. Aumann, winner of the 2008 Nobel Prize in Economics: Artstein, Zvi (1995). «22 Discrete and continuous bang–bang and facial spaces or: Look for the extreme points». Στο: Hart, Sergiu· Neyman, Abraham. Game and economic theory: Selected contributions in honor of Robert J. Aumann. Ann Arbor, MI: University of Michigan Press. σελίδες 449–462. ISBN 0-472-10673-2. Αρχειοθετήθηκε από το πρωτότυπο στις 24 Μαΐου 2011. Ανακτήθηκε στις 25 Ιουνίου 2014.

wikisource.org

en.wikisource.org

Samuelson (1950, pp. 359–360):
It will be noted that any point where the indifference curves are convex rather than concave cannot be observed in a competitive market. Such points are shrouded in eternal darkness—unless we make our consumer a monopsonist and let him choose between goods lying on a very convex "budget curve" (along which he is affecting the price of what he buys). In this monopsony case, we could still deduce the slope of the man's indifference curve from the slope of the observed constraint at the equilibrium point.
Samuelson, Paul A. (Νοέμβριος 1950). «The problem of integrability in utility theory». Economica. New Series 17 (68): 355–385. MR 43436. "Eternal darkness" describes the Hell of John Milton's Paradise Lost, whose concavity is compared to the Serbonian Bog in Book II, lines 592–594:
A gulf profound as that Serbonian Bog

Betwixt Damiata and Mount Casius old,

Where Armies whole have sunk.
Milton's description of concavity serves as the literary epigraph prefacing chapter seven of Arrow & Hahn (1971, p. 169), "Markets with non-convex preferences and production", which presents the results of Starr (1969).

worldcat.org

Schneider (1993, p. 140) credits this result to Borwein & O'Brien (1978): Borwein, Jonathan M.; O'Brien, R. C. (1978). «Cancellation characterizes convexity». Nanta Mathematica (Nanyang University) 11: 100–102. ISSN 0077-2739. MR 510842. Schneider, Rolf (1993). Convex bodies: The Brunn–Minkowski theory. Encyclopedia of Mathematics and its Applications. 44. Cambridge: Cambridge University Press. σελίδες xiv+490. ISBN 0-521-35220-7. MR 1216521.
Ekeland, Ivar (1974). «Une estimation a priori en programmation non convexe» (στα γαλλικά). Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries A et B 279: 149–151. ISSN 0151-0509. MR 395844.

yale.edu

cowles.econ.yale.edu

Howe (1979, p. 1): Howe, Roger (1979-11-03), On the tendency toward convexity of the vector sum of sets, Cowles Foundation discussion papers, 538, Box 2125 Yale Station, New Haven,CT 06520: Cowles Foundation for Research in Economics, Yale University, http://cowles.econ.yale.edu/P/cd/d05a/d0538.pdf, ανακτήθηκε στις 2011-01-01

zentralblatt-math.org

Artstein & Vitale (1975, pp. 881–882): Artstein, Zvi; Vitale, Richard A. (1975), «A strong law of large numbers for random compact sets», The Annals of Probability 3 (5): 879–882, doi:10.1214/aop/1176996275, Πρότυπο:Euclid, Zbl 0313.60012, http://projecteuclid.org/euclid.aop/1176996275
Shapley & Shubik (1966, p. 806): Shapley, L. S.; Shubik, M. (Οκτώβριος 1966). «Quasi-cores in a monetary economy with nonconvex preferences». Econometrica 34 (4): 805–827. doi:10.2307/1910101. Zbl 154.45303. https://archive.org/details/sim_econometrica_1966-10_34_4/page/805.