Shapley–Folkman lemma (English Wikipedia)

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"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 (1980, p. 169), "Markets with non-convex preferences and production", which presents the results of Starr (1969). Arrow, Kenneth J.; Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. Vol. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 ed.). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057. Starr, Ross M. (1969). "Quasi-equilibria in markets with non-convex preferences". Econometrica. 37 (1): 25–38. doi:10.2307/1909201. JSTOR 1909201. Appendix 2: The Shapley–Folkman theorem, pp. 35–37.
Ekeland (1999, pp. 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". In Ekeland, Ivar; Temam, Roger (eds.). Convex analysis and variational problems. Classics in Applied Mathematics. Vol. 28 (Corrected reprinting of the North-Holland ed.). Philadelphia: Society for Industrial and Applied Mathematics (SIAM). pp. 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 (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 et al. (1983)
Bertsekas, Dimitri P. (1996). "5.6 Large scale separable integer programming problems and the exponential method of multipliers". Constrained optimization and Lagrange multiplier methods. Belmont, Mass.: Athena Scientific. ISBN 1-886529-04-3. MR 0690767. Reprint of (1982) Academic Press. Ekeland, Ivar (1999) [1976]. "Appendix I: An a priori estimate in convex programming". In Ekeland, Ivar; Temam, Roger (eds.). Convex analysis and variational problems. Classics in Applied Mathematics. Vol. 28 (Corrected reprinting of the North-Holland ed.). Philadelphia: Society for Industrial and Applied Mathematics (SIAM). pp. 357–373. ISBN 0-89871-450-8. MR 1727362. 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. JSTOR 3689565. MR 0449695. Bertsekas, Dimitri P. (1996). "5.6 Large scale separable integer programming problems and the exponential method of multipliers". Constrained optimization and Lagrange multiplier methods. Belmont, Mass.: Athena Scientific. ISBN 1-886529-04-3. MR 0690767. Reprint of (1982) Academic Press. Bertsekas, Dimitri P.; Lauer, Gregory S.; Sandell, Nils R. Jr.; Posbergh, Thomas A. (January 1983). "Optimal short-term scheduling of large-scale power systems" (PDF). IEEE Transactions on Automatic Control. 28 (1): 1–11. doi:10.1109/tac.1983.1103136. Retrieved 2 February 2011. Proceedings of 1981 IEEE Conference on Decision and Control, San Diego, CA, December 1981, pp. 432–443.
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. JSTOR 2959130. MR 0385966. Zbl 0313.60012. PE euclid.ss/1176996275.
Rockafellar (1997), p. 12. Rockafellar, R. Tyrrell (1997). Convex Analysis. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press. ISBN 0-691-01586-4. MR 1451876.. Reprint of the 1970 (MR 0274683) Princeton Mathematical Series 28
Schneider (1993), pp. 126–196, Chapter 3: Minkowski addition. Schneider, Rolf (1993). Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and its Applications. Vol. 44. Cambridge, UK: Cambridge University Press. ISBN 0-521-35220-7. MR 1216521.
Carter (2001), p. 94. Carter, Michael (2001). Foundations of Mathematical Economics. Cambridge, Mass.: MIT Press. ISBN 0-262-53192-5. MR 1865841.
Arrow & Hahn (1980, p. 375) Arrow, Kenneth J.; Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. Vol. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 ed.). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057.
Rockafellar (1997), p. 10. Rockafellar, R. Tyrrell (1997). Convex Analysis. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press. ISBN 0-691-01586-4. MR 1451876.. Reprint of the 1970 (MR 0274683) Princeton Mathematical Series 28
Blumson (2019). Blumson, Ben (2019). "Naturalness and convex class nominalism". Dialectica. 73 (1–2): 65–81. doi:10.1111/1746-8361.12263. MR 3994693.
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. Vol. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 ed.). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057. Rockafellar, R. Tyrrell (1997). Convex Analysis. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press. ISBN 0-691-01586-4. MR 1451876.. Reprint of the 1970 (MR 0274683) Princeton Mathematical Series 28 Green, Jerry; Heller, Walter P. (1981). "1 Mathematical analysis and convexity with applications to economics". In Arrow, Kenneth Joseph; Intriligator, Michael D. (eds.). Handbook of mathematical economics. Handbooks in Economics. Vol. 1. Amsterdam: North-Holland Publishing. pp. 15–52. doi:10.1016/S1573-4382(81)01005-9. ISBN 0-444-86126-2. MR 0634800.
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. Vol. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 ed.). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057. Rockafellar, R. Tyrrell (1997). Convex Analysis. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press. ISBN 0-691-01586-4. MR 1451876.. Reprint of the 1970 (MR 0274683) Princeton Mathematical Series 28
Schneider (1993, p. xi) and Rockafellar (1997, p. 16) Schneider, Rolf (1993). Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and its Applications. Vol. 44. Cambridge, UK: Cambridge University Press. ISBN 0-521-35220-7. MR 1216521. Rockafellar, R. Tyrrell (1997). Convex Analysis. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press. ISBN 0-691-01586-4. MR 1451876.. Reprint of the 1970 (MR 0274683) Princeton Mathematical Series 28
Rockafellar (1997, p. 17) and Starr (1997, p. 78) Rockafellar, R. Tyrrell (1997). Convex Analysis. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press. ISBN 0-691-01586-4. MR 1451876.. Reprint of the 1970 (MR 0274683) Princeton Mathematical Series 28 Starr, Ross M. (1997). "8 Convex sets, separation theorems, and non-convex sets in R^$N$". General Equilibrium Theory: An Introduction (1st ed.). Cambridge, UK: Cambridge University Press. ISBN 0-521-56473-5. MR 1462618. New chapters 22 and 25–26 in (2011) second ed.
Schneider (1993, pp. 2–3) Schneider, Rolf (1993). Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and its Applications. Vol. 44. Cambridge, UK: Cambridge University Press. 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. Vol. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 ed.). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057.
Carter (2001), p. 93. Carter, Michael (2001). Foundations of Mathematical Economics. Cambridge, Mass.: MIT Press. ISBN 0-262-53192-5. MR 1865841.
Schneider (1993, p. 140) credits this result to Borwein & O'Brien (1978) Schneider, Rolf (1993). Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and its Applications. Vol. 44. Cambridge, UK: Cambridge University Press. ISBN 0-521-35220-7. MR 1216521. Borwein, J. M.; O'Brien, R. C. (1978). "Cancellation characterizes convexity". Nanta Mathematica (Nanyang University). 11: 100–102. ISSN 0077-2739. MR 0510842.
Balestro, Martini & Teixeira (2024), p. 151. Balestro, Vitor; Martini, Horst; Teixeira, Ralph (2024). Convexity from the Geometric Point of View. Cornerstones. Cham: Birkhäuser/Springer. doi:10.1007/978-3-031-50507-2. ISBN 978-3-031-50506-5. MR 4830872.
Schneider (1993, p. 129) Schneider, Rolf (1993). Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and its Applications. Vol. 44. Cambridge, UK: Cambridge University Press. ISBN 0-521-35220-7. MR 1216521.
Starr (1981). Starr, Ross M. (October 1981). "Approximation of points of the convex hull of a sum of sets by points of the sum: An elementary approach" (PDF). Journal of Economic Theory. 25 (2): 314–317. doi:10.1016/0022-0531(81)90010-7. MR 0640201.
Schneider (1993, pp. 129–130) Schneider, Rolf (1993). Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and its Applications. Vol. 44. Cambridge, UK: Cambridge University Press. 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. Vol. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 ed.). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057.
Cassels (1975, pp. 435–436) 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 0385711.
Schneider (1993, p. 128) Schneider, Rolf (1993). Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and its Applications. Vol. 44. Cambridge, UK: Cambridge University Press. 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". In Ekeland, Ivar; Temam, Roger (eds.). Convex analysis and variational problems. Classics in Applied Mathematics. Vol. 28 (Corrected reprinting of the North-Holland ed.). Philadelphia: Society for Industrial and Applied Mathematics (SIAM). pp. 357–373. ISBN 0-89871-450-8. MR 1727362.
Artstein (1980, p. 180) Artstein, Zvi (1980). "Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points". SIAM Review. 22 (2): 172–185. doi:10.1137/1022026. JSTOR 2029960. MR 0564562. Republished in a festschrift for Robert J. Aumann, winner of the 2008 Nobel Prize in Economics:
- Guesnerie (1989, p. 138) Guesnerie, Roger (1989). "First-best allocation of resources with nonconvexities in production". In Cornet, Bernard; Tulkens, Henry (eds.). Contributions to Operations Research and Economics: The twentieth anniversary of CORE (Papers from the symposium held in Louvain-la-Neuve, January 1987). Cambridge, Mass.: MIT Press. pp. 99–143. ISBN 0-262-03149-3. MR 1104662.
- Mas-Colell (1985, pp. 58–61) and Arrow & Hahn (1980, pp. 76–79) Mas-Colell, Andreu (1985). "1.L Averages of sets". The Theory of General Economic Equilibrium: A Differentiable Approach. Econometric Society monographs. Vol. 9. Cambridge University Press. ISBN 0-521-26514-2. MR 1113262. Arrow, Kenneth J.; Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. Vol. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 ed.). 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. Vol. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 ed.). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057.
- Wold (1943b, pp. 231 and 239–240) and Wold (1953, p. 146) Wold, Herman (1943b). "A synthesis of pure demand analysis II". Skandinavisk Aktuarietidskrift [Scandinavian Actuarial Journal]. 26: 220–263. doi:10.1080/03461238.1943.10404737. MR 0011939. Wold, Herman (1953). "8 Some further applications of preference fields (pp. 129–148)". Demand Analysis: A Study in Econometrics. Wiley publications in statistics. In association with Lars Juréen. New York: John Wiley and Sons. 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. (November 1950). "The problem of integrability in utility theory". Economica. New Series. 17 (68): 355–385. doi:10.2307/2549499. JSTOR 2549499. MR 0043436.
- Diewert (1982, pp. 552–553) Diewert, W. E. (1982). "12: Duality approaches to microeconomic theory". In Arrow, Kenneth Joseph; Intriligator, Michael D. (eds.). Handbook of Mathematical Economics. Handbooks in Economics. Vol. 2. Amsterdam: North-Holland Publishing. pp. 535–599. doi:10.1016/S1573-4382(82)02007-4. ISBN 978-0-444-86127-6. MR 0648778.
- Arrow & Hahn (1980, p. 182) Arrow, Kenneth J.; Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. Vol. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 ed.). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057.
- Aumann (1966, pp. 1–2) uses results from Aumann (1964, 1965) Aumann, Robert J. (January 1966). "Existence of competitive equilibrium in markets with a continuum of traders". Econometrica. 34 (1): 1–17. doi:10.2307/1909854. JSTOR 1909854. MR 0191623. Aumann, Robert J. (January–April 1964). "Markets with a continuum of traders". Econometrica. 32 (1–2): 39–50. doi:10.2307/1913732. JSTOR 1913732. MR 0172689. Aumann, Robert J. (August 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 0185073.
- Taking the convex hull of non-convex preferences had been discussed earlier by Wold (1943b, p. 243) and by Wold (1953, p. 146), according to Diewert (1982, p. 552). Wold, Herman (1943b). "A synthesis of pure demand analysis II". Skandinavisk Aktuarietidskrift [Scandinavian Actuarial Journal]. 26: 220–263. doi:10.1080/03461238.1943.10404737. MR 0011939. Wold, Herman (1953). "8 Some further applications of preference fields (pp. 129–148)". Demand Analysis: A Study in Econometrics. Wiley publications in statistics. In association with Lars Juréen. New York: John Wiley and Sons. MR 0064385. Diewert, W. E. (1982). "12: Duality approaches to microeconomic theory". In Arrow, Kenneth Joseph; Intriligator, Michael D. (eds.). Handbook of Mathematical Economics. Handbooks in Economics. Vol. 2. Amsterdam: North-Holland Publishing. pp. 535–599. doi:10.1016/S1573-4382(82)02007-4. ISBN 978-0-444-86127-6. MR 0648778.
- Arrow & Hahn (1980, pp. 169–182) and Starr (1969, pp. 27–33) Arrow, Kenneth J.; Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. Vol. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 ed.). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057. Starr, Ross M. (1969). "Quasi-equilibria in markets with non-convex preferences". Econometrica. 37 (1): 25–38. doi:10.2307/1909201. JSTOR 1909201. Appendix 2: The Shapley–Folkman theorem, pp. 35–37.
- Green & Heller (1981, p. 44) Green, Jerry; Heller, Walter P. (1981). "1 Mathematical analysis and convexity with applications to economics". In Arrow, Kenneth Joseph; Intriligator, Michael D. (eds.). Handbook of mathematical economics. Handbooks in Economics. Vol. 1. Amsterdam: North-Holland Publishing. pp. 15–52. doi:10.1016/S1573-4382(81)01005-9. ISBN 0-444-86126-2. MR 0634800.
- Guesnerie (1989, pp. 99) Guesnerie, Roger (1989). "First-best allocation of resources with nonconvexities in production". In Cornet, Bernard; Tulkens, Henry (eds.). Contributions to Operations Research and Economics: The twentieth anniversary of CORE (Papers from the symposium held in Louvain-la-Neuve, January 1987). Cambridge, Mass.: MIT Press. pp. 99–143. ISBN 0-262-03149-3. MR 1104662.
- Varian (1992, pp. 393–394)
  Mas-Colell, Whinston & Green (1995, pp. 627–630)
  Varian, Hal R. (1992). "21.2 Convexity and size". Microeconomic Analysis (3rd ed.). W. W. Norton & Company. ISBN 978-0-393-95735-8. MR 1036734. 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)
  Hildenbrand (1974, pp. 37, 115–116, 122, and 168)
  Starr (1997, p. 169)
  Ellickson (1994, pp. xviii, 306–310, 312, 328–329, 347, and 352)
  Arrow, Kenneth J.; Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. Vol. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 ed.). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057. Mas-Colell, Andreu (1985). "1.L Averages of sets". The Theory of General Economic Equilibrium: A Differentiable Approach. Econometric Society monographs. Vol. 9. Cambridge University Press. ISBN 0-521-26514-2. MR 1113262. Hildenbrand, Werner (1974). Core and equilibria of a large economy. Princeton studies in mathematical economics. Vol. 5. Princeton, NJ: Princeton University Press. ISBN 978-0-691-04189-6. MR 0389160. Starr, Ross M. (1997). "8 Convex sets, separation theorems, and non-convex sets in R^$N$". General Equilibrium Theory: An Introduction (1st ed.). Cambridge, UK: Cambridge University Press. ISBN 0-521-56473-5. MR 1462618. New chapters 22 and 25–26 in (2011) second ed. Ellickson, Bryan (1994). Competitive equilibrium: Theory and applications. Cambridge University Press. doi:10.1017/CBO9780511609411. 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 [Harcourt Brace Jovanovich, Publishers]. 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. Vol. 62. Cambridge, UK: Cambridge University Press. 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 ed.). Mineola, NY: Dover Publications. 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, Mass.: MIT Press. ISBN 0-262-53192-5. MR 1865841.
- Trockel (1984, p. 30) Trockel, Walter (1984). Market Demand: An Analysis of Large Economies with Nonconvex Preferences. Lecture Notes in Economics and Mathematical Systems. Vol. 223. Berlin: Springer-Verlag. ISBN 3-540-12881-6. MR 0737006.
- Rockafellar (1997, p. 23) Rockafellar, R. Tyrrell (1997). Convex Analysis. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press. ISBN 0-691-01586-4. MR 1451876.. Reprint of the 1970 (MR 0274683) Princeton Mathematical Series 28
- Rockafellar (1997), pp. 49 and 75. Rockafellar, R. Tyrrell (1997). Convex Analysis. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press. ISBN 0-691-01586-4. MR 1451876.. Reprint of the 1970 (MR 0274683) Princeton Mathematical Series 28
- Lemaréchal (1973, p. 38) Lemaréchal's experiments were discussed in later publications:
  Aardal (1995, pp. 2–3)
  Hiriart-Urruty & Lemaréchal (1993, pp. 143–145, 151, 153, and 156)
  Lemaréchal, Claude (April 1973). Utilisation de la dualité dans les problémes non convexes [Use of duality for non–convex problems] (Report) (in French). Domaine de Voluceau, Rocquencourt, Le Chesnay, France: IRIA (now INRIA), Laboratoire de recherche en informatique et automatique. 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]. Vol. 306. Berlin: Springer-Verlag. pp. 136–193 (and bibliographical comments on pp. 334–335). ISBN 3-540-56852-2. MR 1295240.
- Ekeland (1974) Ekeland, Ivar (1974). "Une estimation a priori en programmation non convexe". Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries A et B (in French). 279: 149–151. ISSN 0151-0509. MR 0395844.
- 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. JSTOR 3689565. MR 0449695.
- Aubin & Ekeland (1976) and Ekeland (1999, pp. 362–364) 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. JSTOR 3689565. MR 0449695. Ekeland, Ivar (1999) [1976]. "Appendix I: An a priori estimate in convex programming". In Ekeland, Ivar; Temam, Roger (eds.). Convex analysis and variational problems. Classics in Applied Mathematics. Vol. 28 (Corrected reprinting of the North-Holland ed.). Philadelphia: Society for Industrial and Applied Mathematics (SIAM). pp. 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.
- 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 0385711.
- 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. pp. 194–240. doi:10.1007/1-84628-150-4. ISBN 978-1-84996-949-9. MR 2132405.
- 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. Bibcode:1985MPCPS..97..151P. doi:10.1017/S0305004100062691. MR 0764504.
- 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 0663901.
- Cerf (1999, pp. 243–244) uses applications of the Shapley–Folkman lemma from Puri & Ralescu (1985, pp. 154–155). 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. 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. Bibcode:1985MPCPS..97..151P. doi:10.1017/S0305004100062691. MR 0764504.
- 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.
- Artstein (1980, pp. 172–183) Artstein, Zvi (1980). "Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points". SIAM Review. 22 (2): 172–185. doi:10.1137/1022026. JSTOR 2029960. MR 0564562. Republished in a festschrift for Robert J. Aumann, winner of the 2008 Nobel Prize in Economics:
  - 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 0514468.

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

Schneider (1993), pp. 126–196, Chapter 3: Minkowski addition. Schneider, Rolf (1993). Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and its Applications. Vol. 44. Cambridge, UK: Cambridge University Press. ISBN 0-521-35220-7. MR 1216521.
Schneider (1993, p. xi) and Rockafellar (1997, p. 16) Schneider, Rolf (1993). Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and its Applications. Vol. 44. Cambridge, UK: Cambridge University Press. ISBN 0-521-35220-7. MR 1216521. Rockafellar, R. Tyrrell (1997). Convex Analysis. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press. ISBN 0-691-01586-4. MR 1451876.. Reprint of the 1970 (MR 0274683) Princeton Mathematical Series 28
Schneider (1993, pp. 2–3) Schneider, Rolf (1993). Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and its Applications. Vol. 44. Cambridge, UK: Cambridge University Press. ISBN 0-521-35220-7. MR 1216521.
Schneider (1993, p. 140) credits this result to Borwein & O'Brien (1978) Schneider, Rolf (1993). Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and its Applications. Vol. 44. Cambridge, UK: Cambridge University Press. ISBN 0-521-35220-7. MR 1216521. Borwein, J. M.; O'Brien, R. C. (1978). "Cancellation characterizes convexity". Nanta Mathematica (Nanyang University). 11: 100–102. ISSN 0077-2739. MR 0510842.
Schneider (1993, p. 129) Schneider, Rolf (1993). Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and its Applications. Vol. 44. Cambridge, UK: Cambridge University Press. 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. Vol. 44. Cambridge, UK: Cambridge University Press. ISBN 0-521-35220-7. MR 1216521.
Schneider (1993, p. 128) Schneider, Rolf (1993). Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and its Applications. Vol. 44. Cambridge, UK: Cambridge University Press. ISBN 0-521-35220-7. MR 1216521.
Varian (1992, pp. 393–394)
Mas-Colell, Whinston & Green (1995, pp. 627–630)
Varian, Hal R. (1992). "21.2 Convexity and size". Microeconomic Analysis (3rd ed.). W. W. Norton & Company. ISBN 978-0-393-95735-8. MR 1036734. 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.
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. pp. 194–240. doi:10.1007/1-84628-150-4. ISBN 978-1-84996-949-9. MR 2132405.

berkeley.edu (Global: 580^th place; English: 462^nd place)

eml.berkeley.edu

Anderson (2005). Anderson, Robert M. (14 March 2005). "1 The Shapley–Folkman theorem" (PDF). Economics 201B: Nonconvex preferences and approximate equilibria. Berkeley, Calif.: Economics Department, University of California, Berkeley. pp. 1–5. Retrieved 3 June 2025.

books.google.com (Global: 3^rd place; English: 3^rd place)

Takayama (1985), p. 20. Takayama, Akira (1985). Mathematical Economics. Cambridge University Press. ISBN 9780521314985.
Boules (2021), p. 83. Boules, Adel N. (2021). Fundamentals of Mathematical Analysis. Oxford University Press. ISBN 9780192639448.
Papadopoulos (2005, pp. 105–110, 4.1 The Hausdorff distance). The formula for Hausdorff distance is simplified here using the assumption that $X\subset Y$ to avoid requiring also that $X\subset Y_{\varepsilon }$ . Papadopoulos, Athanase (2005). Metric Spaces, Convexity and Nonpositive Curvature. IRMA lectures in mathematics and theoretical physics. Vol. 6. European Mathematical Society. ISBN 9783037190104.
Balestro, Martini & Teixeira (2024), p. 151. Balestro, Vitor; Martini, Horst; Teixeira, Ralph (2024). Convexity from the Geometric Point of View. Cornerstones. Cham: Birkhäuser/Springer. doi:10.1007/978-3-031-50507-2. ISBN 978-3-031-50506-5. MR 4830872.

dictionaryofeconomics.com (Global: 7,008^th place; English: 5,296^th place)

Starr (2008) Starr, Ross M. (2008). "Shapley–Folkman theorem". In Durlauf, Steven N.; Blume, Lawrence E. (eds.). The new Palgrave Dictionary of Economics (Second ed.). Palgrave Macmillan. pp. 317–318 (1st ed.). doi:10.1057/9780230226203.1518. ISBN 978-0-333-78676-5.
Mas-Colell (1987) Mas-Colell, A. (1987). "Non-convexity". In Eatwell, John; Milgate, Murray; Newman, Peter (eds.). The New Palgrave: A Dictionary of Economics (first ed.). Palgrave Macmillan. pp. 653–661. doi:10.1057/9780230226203.3173. ISBN 9780333786765. (PDF file at Mas-Colell's homepage).

doi.org (Global: 2^nd place; English: 2^nd place)

"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 (1980, p. 169), "Markets with non-convex preferences and production", which presents the results of Starr (1969). Arrow, Kenneth J.; Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. Vol. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 ed.). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057. Starr, Ross M. (1969). "Quasi-equilibria in markets with non-convex preferences". Econometrica. 37 (1): 25–38. doi:10.2307/1909201. JSTOR 1909201. Appendix 2: The Shapley–Folkman theorem, pp. 35–37.
Starr (1969) Starr, Ross M. (1969). "Quasi-equilibria in markets with non-convex preferences". Econometrica. 37 (1): 25–38. doi:10.2307/1909201. JSTOR 1909201. Appendix 2: The Shapley–Folkman theorem, pp. 35–37.
Starr (2008) Starr, Ross M. (2008). "Shapley–Folkman theorem". In Durlauf, Steven N.; Blume, Lawrence E. (eds.). The new Palgrave Dictionary of Economics (Second ed.). Palgrave Macmillan. pp. 317–318 (1st ed.). doi:10.1057/9780230226203.1518. ISBN 978-0-333-78676-5.
Bertsekas (1996, pp. 364–381) acknowledging Ekeland (1999) on page 374 and Aubin & Ekeland (1976) on page 381:
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 et al. (1983)
Bertsekas, Dimitri P. (1996). "5.6 Large scale separable integer programming problems and the exponential method of multipliers". Constrained optimization and Lagrange multiplier methods. Belmont, Mass.: Athena Scientific. ISBN 1-886529-04-3. MR 0690767. Reprint of (1982) Academic Press. Ekeland, Ivar (1999) [1976]. "Appendix I: An a priori estimate in convex programming". In Ekeland, Ivar; Temam, Roger (eds.). Convex analysis and variational problems. Classics in Applied Mathematics. Vol. 28 (Corrected reprinting of the North-Holland ed.). Philadelphia: Society for Industrial and Applied Mathematics (SIAM). pp. 357–373. ISBN 0-89871-450-8. MR 1727362. 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. JSTOR 3689565. MR 0449695. Bertsekas, Dimitri P. (1996). "5.6 Large scale separable integer programming problems and the exponential method of multipliers". Constrained optimization and Lagrange multiplier methods. Belmont, Mass.: Athena Scientific. ISBN 1-886529-04-3. MR 0690767. Reprint of (1982) Academic Press. Bertsekas, Dimitri P.; Lauer, Gregory S.; Sandell, Nils R. Jr.; Posbergh, Thomas A. (January 1983). "Optimal short-term scheduling of large-scale power systems" (PDF). IEEE Transactions on Automatic Control. 28 (1): 1–11. doi:10.1109/tac.1983.1103136. Retrieved 2 February 2011. Proceedings of 1981 IEEE Conference on Decision and Control, San Diego, CA, December 1981, pp. 432–443.
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. JSTOR 2959130. MR 0385966. Zbl 0313.60012. PE euclid.ss/1176996275.
Blumson (2019). Blumson, Ben (2019). "Naturalness and convex class nominalism". Dialectica. 73 (1–2): 65–81. doi:10.1111/1746-8361.12263. MR 3994693.
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. Vol. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 ed.). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057. Rockafellar, R. Tyrrell (1997). Convex Analysis. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press. ISBN 0-691-01586-4. MR 1451876.. Reprint of the 1970 (MR 0274683) Princeton Mathematical Series 28 Green, Jerry; Heller, Walter P. (1981). "1 Mathematical analysis and convexity with applications to economics". In Arrow, Kenneth Joseph; Intriligator, Michael D. (eds.). Handbook of mathematical economics. Handbooks in Economics. Vol. 1. Amsterdam: North-Holland Publishing. pp. 15–52. doi:10.1016/S1573-4382(81)01005-9. ISBN 0-444-86126-2. MR 0634800.
Starr (1969, pp. 35–36) Starr, Ross M. (1969). "Quasi-equilibria in markets with non-convex preferences". Econometrica. 37 (1): 25–38. doi:10.2307/1909201. JSTOR 1909201. Appendix 2: The Shapley–Folkman theorem, pp. 35–37.
Balestro, Martini & Teixeira (2024), p. 151. Balestro, Vitor; Martini, Horst; Teixeira, Ralph (2024). Convexity from the Geometric Point of View. Cornerstones. Cham: Birkhäuser/Springer. doi:10.1007/978-3-031-50507-2. ISBN 978-3-031-50506-5. MR 4830872.
Starr (1969, p. 36) Starr, Ross M. (1969). "Quasi-equilibria in markets with non-convex preferences". Econometrica. 37 (1): 25–38. doi:10.2307/1909201. JSTOR 1909201. Appendix 2: The Shapley–Folkman theorem, pp. 35–37.
Starr (1981). Starr, Ross M. (October 1981). "Approximation of points of the convex hull of a sum of sets by points of the sum: An elementary approach" (PDF). Journal of Economic Theory. 25 (2): 314–317. doi:10.1016/0022-0531(81)90010-7. MR 0640201.
Starr (1969, p. 37) Starr, Ross M. (1969). "Quasi-equilibria in markets with non-convex preferences". Econometrica. 37 (1): 25–38. doi:10.2307/1909201. JSTOR 1909201. Appendix 2: The Shapley–Folkman theorem, pp. 35–37.
Cassels (1975, pp. 435–436) 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 0385711.
Artstein (1980, p. 180) Artstein, Zvi (1980). "Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points". SIAM Review. 22 (2): 172–185. doi:10.1137/1022026. JSTOR 2029960. MR 0564562. Republished in a festschrift for Robert J. Aumann, winner of the 2008 Nobel Prize in Economics:
- Zhou (1993). Zhou, Lin (June 1993). "A simple proof of the Shapley–Folkman theorem". Economic Theory. 3 (2): 371–372. doi:10.1007/bf01212924. ISSN 0938-2259.
- Starr (1969, p. 26): "After all, one may be indifferent between an automobile and a boat, but in most cases one can neither drive nor sail the combination of half boat, half car." Starr, Ross M. (1969). "Quasi-equilibria in markets with non-convex preferences". Econometrica. 37 (1): 25–38. doi:10.2307/1909201. JSTOR 1909201. Appendix 2: The Shapley–Folkman theorem, pp. 35–37.
- Hotelling (1935, p. 74) Hotelling, Harold (January 1935). "Demand functions with limited budgets". Econometrica. 3 (1): 66–78. doi:10.2307/1907346. JSTOR 1907346.
- Wold (1943b, pp. 231 and 239–240) and Wold (1953, p. 146) Wold, Herman (1943b). "A synthesis of pure demand analysis II". Skandinavisk Aktuarietidskrift [Scandinavian Actuarial Journal]. 26: 220–263. doi:10.1080/03461238.1943.10404737. MR 0011939. Wold, Herman (1953). "8 Some further applications of preference fields (pp. 129–148)". Demand Analysis: A Study in Econometrics. Wiley publications in statistics. In association with Lars Juréen. New York: John Wiley and Sons. 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. (November 1950). "The problem of integrability in utility theory". Economica. New Series. 17 (68): 355–385. doi:10.2307/2549499. JSTOR 2549499. MR 0043436.
- Diewert (1982, pp. 552–553) Diewert, W. E. (1982). "12: Duality approaches to microeconomic theory". In Arrow, Kenneth Joseph; Intriligator, Michael D. (eds.). Handbook of Mathematical Economics. Handbooks in Economics. Vol. 2. Amsterdam: North-Holland Publishing. pp. 535–599. doi:10.1016/S1573-4382(82)02007-4. ISBN 978-0-444-86127-6. MR 0648778.
- Farrell (1959, 1961a, 1961b) Farrell, M. J. (August 1959). "The Convexity assumption in the theory of competitive markets". Journal of Political Economy. 67 (4): 371–391. doi:10.1086/258197. JSTOR 1825163. Farrell, M. J. (October 1961a). "On Convexity, efficiency, and markets: A Reply". Journal of Political Economy. 69 (5): 484–489. doi:10.1086/258541. JSTOR 1828538. Farrell, M. J. (October 1961b). "The Convexity assumption in the theory of competitive markets: Rejoinder". Journal of Political Economy. 69 (5): 493. doi:10.1086/258544. JSTOR 1828541.
- Bator (1961a, 1961b) Bator, Francis M. (October 1961a). "On convexity, efficiency, and markets". Journal of Political Economy. 69 (5): 480–483. doi:10.1086/258540. JSTOR 1828537.
  - Koopmans (1961, p. 478) and others—for example, Farrell (1959, pp. 390–391) and Farrell (1961a, p. 484), Bator (1961a, 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]) Koopmans, Tjalling C. (October 1961). "Convexity assumptions, allocative efficiency, and competitive equilibrium". Journal of Political Economy. 69 (5): 478–479. doi:10.1086/258539. JSTOR 1828536. Farrell, M. J. (August 1959). "The Convexity assumption in the theory of competitive markets". Journal of Political Economy. 67 (4): 371–391. doi:10.1086/258197. JSTOR 1825163. Farrell, M. J. (October 1961a). "On Convexity, efficiency, and markets: A Reply". Journal of Political Economy. 69 (5): 484–489. doi:10.1086/258541. JSTOR 1828538. Bator, Francis M. (October 1961a). "On convexity, efficiency, and markets". Journal of Political Economy. 69 (5): 480–483. doi:10.1086/258540. JSTOR 1828537.
    - Rothenberg (1960, p. 447, 1961) Rothenberg, Jerome (October 1960). "Non-convexity, aggregation, and Pareto optimality". Journal of Political Economy. 68 (5): 435–468. doi:10.1086/258363. JSTOR 1830308. Rothenberg, Jerome (October 1961). "Comments on non-convexity". Journal of Political Economy. 69 (5): 490–492. doi:10.1086/258543. JSTOR 1828540.
    - Shapley & Shubik (1966, p. 806) Shapley, L. S.; Shubik, M. (October 1966). "Quasi-cores in a monetary economy with nonconvex preferences". Econometrica. 34 (4): 805–827. doi:10.2307/1910101. JSTOR 1910101. Zbl 0154.45303. Archived from the original on 24 September 2017.
    - Aumann (1966, pp. 1–2) uses results from Aumann (1964, 1965) Aumann, Robert J. (January 1966). "Existence of competitive equilibrium in markets with a continuum of traders". Econometrica. 34 (1): 1–17. doi:10.2307/1909854. JSTOR 1909854. MR 0191623. Aumann, Robert J. (January–April 1964). "Markets with a continuum of traders". Econometrica. 32 (1–2): 39–50. doi:10.2307/1913732. JSTOR 1913732. MR 0172689. Aumann, Robert J. (August 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 0185073.
    - Taking the convex hull of non-convex preferences had been discussed earlier by Wold (1943b, p. 243) and by Wold (1953, p. 146), according to Diewert (1982, p. 552). Wold, Herman (1943b). "A synthesis of pure demand analysis II". Skandinavisk Aktuarietidskrift [Scandinavian Actuarial Journal]. 26: 220–263. doi:10.1080/03461238.1943.10404737. MR 0011939. Wold, Herman (1953). "8 Some further applications of preference fields (pp. 129–148)". Demand Analysis: A Study in Econometrics. Wiley publications in statistics. In association with Lars Juréen. New York: John Wiley and Sons. MR 0064385. Diewert, W. E. (1982). "12: Duality approaches to microeconomic theory". In Arrow, Kenneth Joseph; Intriligator, Michael D. (eds.). Handbook of Mathematical Economics. Handbooks in Economics. Vol. 2. Amsterdam: North-Holland Publishing. pp. 535–599. doi:10.1016/S1573-4382(82)02007-4. ISBN 978-0-444-86127-6. MR 0648778.
    - Starr & Stinchcombe (1999, pp. 217–218) Starr, Ross M.; Stinchcombe, M. B. (1999). "Exchange in a network of trading posts". In Chichilnisky, Graciela (ed.). Markets, Information and Uncertainty: Essays in Economic Theory in Honor of Kenneth J. Arrow. Cambridge, UK: Cambridge University Press. pp. 217–234. doi:10.1017/CBO9780511896583. ISBN 978-0-521-08288-4.
    - Arrow & Hahn (1980, pp. 169–182) and Starr (1969, pp. 27–33) Arrow, Kenneth J.; Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. Vol. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 ed.). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057. Starr, Ross M. (1969). "Quasi-equilibria in markets with non-convex preferences". Econometrica. 37 (1): 25–38. doi:10.2307/1909201. JSTOR 1909201. Appendix 2: The Shapley–Folkman theorem, pp. 35–37.
    - Green & Heller (1981, p. 44) Green, Jerry; Heller, Walter P. (1981). "1 Mathematical analysis and convexity with applications to economics". In Arrow, Kenneth Joseph; Intriligator, Michael D. (eds.). Handbook of mathematical economics. Handbooks in Economics. Vol. 1. Amsterdam: North-Holland Publishing. pp. 15–52. doi:10.1016/S1573-4382(81)01005-9. ISBN 0-444-86126-2. MR 0634800.
    - Mas-Colell (1987) Mas-Colell, A. (1987). "Non-convexity". In Eatwell, John; Milgate, Murray; Newman, Peter (eds.). The New Palgrave: A Dictionary of Economics (first ed.). Palgrave Macmillan. pp. 653–661. doi:10.1057/9780230226203.3173. ISBN 9780333786765. (PDF file at Mas-Colell's homepage).
    - Arrow & Hahn (1980, pp. 169–182)
      Mas-Colell (1985, pp. 52–55, 145–146, 152–153, and 274–275)
      Hildenbrand (1974, pp. 37, 115–116, 122, and 168)
      Starr (1997, p. 169)
      Ellickson (1994, pp. xviii, 306–310, 312, 328–329, 347, and 352)
      Arrow, Kenneth J.; Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. Vol. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 ed.). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057. Mas-Colell, Andreu (1985). "1.L Averages of sets". The Theory of General Economic Equilibrium: A Differentiable Approach. Econometric Society monographs. Vol. 9. Cambridge University Press. ISBN 0-521-26514-2. MR 1113262. Hildenbrand, Werner (1974). Core and equilibria of a large economy. Princeton studies in mathematical economics. Vol. 5. Princeton, NJ: Princeton University Press. ISBN 978-0-691-04189-6. MR 0389160. Starr, Ross M. (1997). "8 Convex sets, separation theorems, and non-convex sets in R^$N$". General Equilibrium Theory: An Introduction (1st ed.). Cambridge, UK: Cambridge University Press. ISBN 0-521-56473-5. MR 1462618. New chapters 22 and 25–26 in (2011) second ed. Ellickson, Bryan (1994). Competitive equilibrium: Theory and applications. Cambridge University Press. doi:10.1017/CBO9780511609411. ISBN 978-0-521-31988-1.
    - 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. JSTOR 3689565. MR 0449695.
    - Aubin & Ekeland (1976) and Ekeland (1999, pp. 362–364) 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. JSTOR 3689565. MR 0449695. Ekeland, Ivar (1999) [1976]. "Appendix I: An a priori estimate in convex programming". In Ekeland, Ivar; Temam, Roger (eds.). Convex analysis and variational problems. Classics in Applied Mathematics. Vol. 28 (Corrected reprinting of the North-Holland ed.). Philadelphia: Society for Industrial and Applied Mathematics (SIAM). pp. 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.
    - 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 0385711.
    - 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. pp. 194–240. doi:10.1007/1-84628-150-4. ISBN 978-1-84996-949-9. MR 2132405.
    - 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. Bibcode:1985MPCPS..97..151P. doi:10.1017/S0305004100062691. MR 0764504.
    - 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 0663901.
    - Cerf (1999, pp. 243–244) uses applications of the Shapley–Folkman lemma from Puri & Ralescu (1985, pp. 154–155). 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. 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. Bibcode:1985MPCPS..97..151P. doi:10.1017/S0305004100062691. MR 0764504.
    - 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.
    - Vind (1964, pp. 168 and 175) 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]
      
      Vind, Karl (May 1964). "Edgeworth-allocations in an exchange economy with many traders". International Economic Review. 5 (2): 165–177. doi:10.2307/2525560. JSTOR 2525560. Debreu, Gérard (March 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. JSTOR 2006785. Vind, Karl (May 1964). "Edgeworth-allocations in an exchange economy with many traders". International Economic Review. 5 (2): 165–177. doi:10.2307/2525560. JSTOR 2525560.
    - Artstein (1980, pp. 172–183) Artstein, Zvi (1980). "Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points". SIAM Review. 22 (2): 172–185. doi:10.1137/1022026. JSTOR 2029960. MR 0564562. Republished in a festschrift for Robert J. Aumann, winner of the 2008 Nobel Prize in Economics:
      - 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 0514468.

dtic.mil (Global: 833^rd place; English: 567^th place)

Shapley & Shubik (1966, p. 806) Shapley, L. S.; Shubik, M. (October 1966). "Quasi-cores in a monetary economy with nonconvex preferences". Econometrica. 34 (4): 805–827. doi:10.2307/1910101. JSTOR 1910101. Zbl 0154.45303. Archived from the original on 24 September 2017.

harvard.edu (Global: 18^th place; English: 17^th place)

ui.adsabs.harvard.edu

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. Bibcode:1985MPCPS..97..151P. doi:10.1017/S0305004100062691. MR 0764504.
Cerf (1999, pp. 243–244) uses applications of the Shapley–Folkman lemma from Puri & Ralescu (1985, pp. 154–155). 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. 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. Bibcode:1985MPCPS..97..151P. doi:10.1017/S0305004100062691. MR 0764504.

jstor.org (Global: 26^th place; English: 20^th place)

"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 (1980, p. 169), "Markets with non-convex preferences and production", which presents the results of Starr (1969). Arrow, Kenneth J.; Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. Vol. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 ed.). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057. Starr, Ross M. (1969). "Quasi-equilibria in markets with non-convex preferences". Econometrica. 37 (1): 25–38. doi:10.2307/1909201. JSTOR 1909201. Appendix 2: The Shapley–Folkman theorem, pp. 35–37.
Starr (1969) Starr, Ross M. (1969). "Quasi-equilibria in markets with non-convex preferences". Econometrica. 37 (1): 25–38. doi:10.2307/1909201. JSTOR 1909201. Appendix 2: The Shapley–Folkman theorem, pp. 35–37.
Bertsekas (1996, pp. 364–381) acknowledging Ekeland (1999) on page 374 and Aubin & Ekeland (1976) on page 381:
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 et al. (1983)
Bertsekas, Dimitri P. (1996). "5.6 Large scale separable integer programming problems and the exponential method of multipliers". Constrained optimization and Lagrange multiplier methods. Belmont, Mass.: Athena Scientific. ISBN 1-886529-04-3. MR 0690767. Reprint of (1982) Academic Press. Ekeland, Ivar (1999) [1976]. "Appendix I: An a priori estimate in convex programming". In Ekeland, Ivar; Temam, Roger (eds.). Convex analysis and variational problems. Classics in Applied Mathematics. Vol. 28 (Corrected reprinting of the North-Holland ed.). Philadelphia: Society for Industrial and Applied Mathematics (SIAM). pp. 357–373. ISBN 0-89871-450-8. MR 1727362. 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. JSTOR 3689565. MR 0449695. Bertsekas, Dimitri P. (1996). "5.6 Large scale separable integer programming problems and the exponential method of multipliers". Constrained optimization and Lagrange multiplier methods. Belmont, Mass.: Athena Scientific. ISBN 1-886529-04-3. MR 0690767. Reprint of (1982) Academic Press. Bertsekas, Dimitri P.; Lauer, Gregory S.; Sandell, Nils R. Jr.; Posbergh, Thomas A. (January 1983). "Optimal short-term scheduling of large-scale power systems" (PDF). IEEE Transactions on Automatic Control. 28 (1): 1–11. doi:10.1109/tac.1983.1103136. Retrieved 2 February 2011. Proceedings of 1981 IEEE Conference on Decision and Control, San Diego, CA, December 1981, pp. 432–443.
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. JSTOR 2959130. MR 0385966. Zbl 0313.60012. PE euclid.ss/1176996275.
Starr (1969, pp. 35–36) Starr, Ross M. (1969). "Quasi-equilibria in markets with non-convex preferences". Econometrica. 37 (1): 25–38. doi:10.2307/1909201. JSTOR 1909201. Appendix 2: The Shapley–Folkman theorem, pp. 35–37.
Starr (1969, p. 36) Starr, Ross M. (1969). "Quasi-equilibria in markets with non-convex preferences". Econometrica. 37 (1): 25–38. doi:10.2307/1909201. JSTOR 1909201. Appendix 2: The Shapley–Folkman theorem, pp. 35–37.
Starr (1969, p. 37) Starr, Ross M. (1969). "Quasi-equilibria in markets with non-convex preferences". Econometrica. 37 (1): 25–38. doi:10.2307/1909201. JSTOR 1909201. Appendix 2: The Shapley–Folkman theorem, pp. 35–37.
Artstein (1980, p. 180) Artstein, Zvi (1980). "Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points". SIAM Review. 22 (2): 172–185. doi:10.1137/1022026. JSTOR 2029960. MR 0564562. Republished in a festschrift for Robert J. Aumann, winner of the 2008 Nobel Prize in Economics:
- Starr (1969, p. 26): "After all, one may be indifferent between an automobile and a boat, but in most cases one can neither drive nor sail the combination of half boat, half car." Starr, Ross M. (1969). "Quasi-equilibria in markets with non-convex preferences". Econometrica. 37 (1): 25–38. doi:10.2307/1909201. JSTOR 1909201. Appendix 2: The Shapley–Folkman theorem, pp. 35–37.
- Hotelling (1935, p. 74) Hotelling, Harold (January 1935). "Demand functions with limited budgets". Econometrica. 3 (1): 66–78. doi:10.2307/1907346. JSTOR 1907346.
- 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. (November 1950). "The problem of integrability in utility theory". Economica. New Series. 17 (68): 355–385. doi:10.2307/2549499. JSTOR 2549499. MR 0043436.
- Farrell (1959, 1961a, 1961b) Farrell, M. J. (August 1959). "The Convexity assumption in the theory of competitive markets". Journal of Political Economy. 67 (4): 371–391. doi:10.1086/258197. JSTOR 1825163. Farrell, M. J. (October 1961a). "On Convexity, efficiency, and markets: A Reply". Journal of Political Economy. 69 (5): 484–489. doi:10.1086/258541. JSTOR 1828538. Farrell, M. J. (October 1961b). "The Convexity assumption in the theory of competitive markets: Rejoinder". Journal of Political Economy. 69 (5): 493. doi:10.1086/258544. JSTOR 1828541.
- Bator (1961a, 1961b) Bator, Francis M. (October 1961a). "On convexity, efficiency, and markets". Journal of Political Economy. 69 (5): 480–483. doi:10.1086/258540. JSTOR 1828537.
  - Koopmans (1961, p. 478) and others—for example, Farrell (1959, pp. 390–391) and Farrell (1961a, p. 484), Bator (1961a, 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]) Koopmans, Tjalling C. (October 1961). "Convexity assumptions, allocative efficiency, and competitive equilibrium". Journal of Political Economy. 69 (5): 478–479. doi:10.1086/258539. JSTOR 1828536. Farrell, M. J. (August 1959). "The Convexity assumption in the theory of competitive markets". Journal of Political Economy. 67 (4): 371–391. doi:10.1086/258197. JSTOR 1825163. Farrell, M. J. (October 1961a). "On Convexity, efficiency, and markets: A Reply". Journal of Political Economy. 69 (5): 484–489. doi:10.1086/258541. JSTOR 1828538. Bator, Francis M. (October 1961a). "On convexity, efficiency, and markets". Journal of Political Economy. 69 (5): 480–483. doi:10.1086/258540. JSTOR 1828537.
    - Rothenberg (1960, p. 447, 1961) Rothenberg, Jerome (October 1960). "Non-convexity, aggregation, and Pareto optimality". Journal of Political Economy. 68 (5): 435–468. doi:10.1086/258363. JSTOR 1830308. Rothenberg, Jerome (October 1961). "Comments on non-convexity". Journal of Political Economy. 69 (5): 490–492. doi:10.1086/258543. JSTOR 1828540.
    - Shapley & Shubik (1966, p. 806) Shapley, L. S.; Shubik, M. (October 1966). "Quasi-cores in a monetary economy with nonconvex preferences". Econometrica. 34 (4): 805–827. doi:10.2307/1910101. JSTOR 1910101. Zbl 0154.45303. Archived from the original on 24 September 2017.
    - Aumann (1966, pp. 1–2) uses results from Aumann (1964, 1965) Aumann, Robert J. (January 1966). "Existence of competitive equilibrium in markets with a continuum of traders". Econometrica. 34 (1): 1–17. doi:10.2307/1909854. JSTOR 1909854. MR 0191623. Aumann, Robert J. (January–April 1964). "Markets with a continuum of traders". Econometrica. 32 (1–2): 39–50. doi:10.2307/1913732. JSTOR 1913732. MR 0172689. Aumann, Robert J. (August 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 0185073.
    - Arrow & Hahn (1980, pp. 169–182) and Starr (1969, pp. 27–33) Arrow, Kenneth J.; Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. Vol. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 ed.). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057. Starr, Ross M. (1969). "Quasi-equilibria in markets with non-convex preferences". Econometrica. 37 (1): 25–38. doi:10.2307/1909201. JSTOR 1909201. Appendix 2: The Shapley–Folkman theorem, pp. 35–37.
    - 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. JSTOR 3689565. MR 0449695.
    - Aubin & Ekeland (1976) and Ekeland (1999, pp. 362–364) 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. JSTOR 3689565. MR 0449695. Ekeland, Ivar (1999) [1976]. "Appendix I: An a priori estimate in convex programming". In Ekeland, Ivar; Temam, Roger (eds.). Convex analysis and variational problems. Classics in Applied Mathematics. Vol. 28 (Corrected reprinting of the North-Holland ed.). Philadelphia: Society for Industrial and Applied Mathematics (SIAM). pp. 357–373. ISBN 0-89871-450-8. MR 1727362.
    - Vind (1964, pp. 168 and 175) 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]
      
      Vind, Karl (May 1964). "Edgeworth-allocations in an exchange economy with many traders". International Economic Review. 5 (2): 165–177. doi:10.2307/2525560. JSTOR 2525560. Debreu, Gérard (March 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. JSTOR 2006785. Vind, Karl (May 1964). "Edgeworth-allocations in an exchange economy with many traders". International Economic Review. 5 (2): 165–177. doi:10.2307/2525560. JSTOR 2525560.
    - Artstein (1980, pp. 172–183) Artstein, Zvi (1980). "Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points". SIAM Review. 22 (2): 172–185. doi:10.1137/1022026. JSTOR 2029960. MR 0564562. Republished in a festschrift for Robert J. Aumann, winner of the 2008 Nobel Prize in Economics:

mit.edu (Global: 415^th place; English: 327^th place)

web.mit.edu

Bertsekas (1996, pp. 364–381) acknowledging Ekeland (1999) on page 374 and Aubin & Ekeland (1976) on page 381:
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 et al. (1983)
Bertsekas, Dimitri P. (1996). "5.6 Large scale separable integer programming problems and the exponential method of multipliers". Constrained optimization and Lagrange multiplier methods. Belmont, Mass.: Athena Scientific. ISBN 1-886529-04-3. MR 0690767. Reprint of (1982) Academic Press. Ekeland, Ivar (1999) [1976]. "Appendix I: An a priori estimate in convex programming". In Ekeland, Ivar; Temam, Roger (eds.). Convex analysis and variational problems. Classics in Applied Mathematics. Vol. 28 (Corrected reprinting of the North-Holland ed.). Philadelphia: Society for Industrial and Applied Mathematics (SIAM). pp. 357–373. ISBN 0-89871-450-8. MR 1727362. 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. JSTOR 3689565. MR 0449695. Bertsekas, Dimitri P. (1996). "5.6 Large scale separable integer programming problems and the exponential method of multipliers". Constrained optimization and Lagrange multiplier methods. Belmont, Mass.: Athena Scientific. ISBN 1-886529-04-3. MR 0690767. Reprint of (1982) Academic Press. Bertsekas, Dimitri P.; Lauer, Gregory S.; Sandell, Nils R. Jr.; Posbergh, Thomas A. (January 1983). "Optimal short-term scheduling of large-scale power systems" (PDF). IEEE Transactions on Automatic Control. 28 (1): 1–11. doi:10.1109/tac.1983.1103136. Retrieved 2 February 2011. Proceedings of 1981 IEEE Conference on Decision and Control, San Diego, CA, December 1981, pp. 432–443.

projecteuclid.org (Global: 3,707^th place; English: 2,409^th place)

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. JSTOR 2959130. MR 0385966. Zbl 0313.60012. PE euclid.ss/1176996275.

sciencedirect.com (Global: 149^th place; English: 178^th place)

Diewert (1982, pp. 552–553) Diewert, W. E. (1982). "12: Duality approaches to microeconomic theory". In Arrow, Kenneth Joseph; Intriligator, Michael D. (eds.). Handbook of Mathematical Economics. Handbooks in Economics. Vol. 2. Amsterdam: North-Holland Publishing. pp. 535–599. doi:10.1016/S1573-4382(82)02007-4. ISBN 978-0-444-86127-6. MR 0648778.
Taking the convex hull of non-convex preferences had been discussed earlier by Wold (1943b, p. 243) and by Wold (1953, p. 146), according to Diewert (1982, p. 552). Wold, Herman (1943b). "A synthesis of pure demand analysis II". Skandinavisk Aktuarietidskrift [Scandinavian Actuarial Journal]. 26: 220–263. doi:10.1080/03461238.1943.10404737. MR 0011939. Wold, Herman (1953). "8 Some further applications of preference fields (pp. 129–148)". Demand Analysis: A Study in Econometrics. Wiley publications in statistics. In association with Lars Juréen. New York: John Wiley and Sons. MR 0064385. Diewert, W. E. (1982). "12: Duality approaches to microeconomic theory". In Arrow, Kenneth Joseph; Intriligator, Michael D. (eds.). Handbook of Mathematical Economics. Handbooks in Economics. Vol. 2. Amsterdam: North-Holland Publishing. pp. 535–599. doi:10.1016/S1573-4382(82)02007-4. ISBN 978-0-444-86127-6. MR 0648778.

ucsd.edu (Global: 1,933^rd place; English: 1,342^nd place)

econweb.ucsd.edu

Starr (1981). Starr, Ross M. (October 1981). "Approximation of points of the convex hull of a sum of sets by points of the sum: An elementary approach" (PDF). Journal of Economic Theory. 25 (2): 314–317. doi:10.1016/0022-0531(81)90010-7. MR 0640201.

umich.edu (Global: 459^th place; English: 360^th place)

press.umich.edu

Artstein (1980, p. 180) Artstein, Zvi (1980). "Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points". SIAM Review. 22 (2): 172–185. doi:10.1137/1022026. JSTOR 2029960. MR 0564562. Republished in a festschrift for Robert J. Aumann, winner of the 2008 Nobel Prize in Economics:
- Artstein (1980, pp. 172–183) Artstein, Zvi (1980). "Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points". SIAM Review. 22 (2): 172–185. doi:10.1137/1022026. JSTOR 2029960. MR 0564562. Republished in a festschrift for Robert J. Aumann, winner of the 2008 Nobel Prize in Economics:

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

econ.upf.edu

Mas-Colell (1987) Mas-Colell, A. (1987). "Non-convexity". In Eatwell, John; Milgate, Murray; Newman, Peter (eds.). The New Palgrave: A Dictionary of Economics (first ed.). Palgrave Macmillan. pp. 653–661. doi:10.1057/9780230226203.3173. ISBN 9780333786765. (PDF file at Mas-Colell's homepage).

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

Artstein (1980, p. 180) Artstein, Zvi (1980). "Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points". SIAM Review. 22 (2): 172–185. doi:10.1137/1022026. JSTOR 2029960. MR 0564562. Republished in a festschrift for Robert J. Aumann, winner of the 2008 Nobel Prize in Economics:
- Shapley & Shubik (1966, p. 806) Shapley, L. S.; Shubik, M. (October 1966). "Quasi-cores in a monetary economy with nonconvex preferences". Econometrica. 34 (4): 805–827. doi:10.2307/1910101. JSTOR 1910101. Zbl 0154.45303. Archived from the original on 24 September 2017.
- Artstein (1980, pp. 172–183) Artstein, Zvi (1980). "Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points". SIAM Review. 22 (2): 172–185. doi:10.1137/1022026. JSTOR 2029960. MR 0564562. Republished in a festschrift for Robert J. Aumann, winner of the 2008 Nobel Prize in Economics:

wikisource.org (Global: 27^th place; English: 51^st place)

en.wikisource.org

"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 (1980, p. 169), "Markets with non-convex preferences and production", which presents the results of Starr (1969). Arrow, Kenneth J.; Hahn, Frank H. (1980) [1971]. General competitive analysis. Advanced Textbooks in Economics. Vol. 12 (reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts 6 ed.). Amsterdam: North-Holland. ISBN 0-444-85497-5. MR 0439057. Starr, Ross M. (1969). "Quasi-equilibria in markets with non-convex preferences". Econometrica. 37 (1): 25–38. doi:10.2307/1909201. JSTOR 1909201. Appendix 2: The Shapley–Folkman theorem, pp. 35–37.

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

search.worldcat.org

Schneider (1993, p. 140) credits this result to Borwein & O'Brien (1978) Schneider, Rolf (1993). Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and its Applications. Vol. 44. Cambridge, UK: Cambridge University Press. ISBN 0-521-35220-7. MR 1216521. Borwein, J. M.; O'Brien, R. C. (1978). "Cancellation characterizes convexity". Nanta Mathematica (Nanyang University). 11: 100–102. ISSN 0077-2739. MR 0510842.
Zhou (1993). Zhou, Lin (June 1993). "A simple proof of the Shapley–Folkman theorem". Economic Theory. 3 (2): 371–372. doi:10.1007/bf01212924. ISSN 0938-2259.
Ekeland (1974) Ekeland, Ivar (1974). "Une estimation a priori en programmation non convexe". Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries A et B (in French). 279: 149–151. ISSN 0151-0509. MR 0395844.

yale.edu (Global: 565^th place; English: 460^th place)

cowles.econ.yale.edu

Howe (1979, p. 1) Howe, Roger (November 1979). On the tendency toward convexity of the vector sum of sets (PDF) (Report). Cowles Foundation discussion papers. Vol. 538. New Haven, Conn.: Cowles Foundation for Research in Economics, Yale University. Retrieved 15 January 2011.

zbmath.org (Global: 1,923^rd place; English: 1,068^th place)

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. JSTOR 2959130. MR 0385966. Zbl 0313.60012. PE euclid.ss/1176996275.
Shapley & Shubik (1966, p. 806) Shapley, L. S.; Shubik, M. (October 1966). "Quasi-cores in a monetary economy with nonconvex preferences". Econometrica. 34 (4): 805–827. doi:10.2307/1910101. JSTOR 1910101. Zbl 0154.45303. Archived from the original on 24 September 2017.

Shapley–Folkman lemma (English Wikipedia)

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