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• Shapley–Folkman_lemma abstract "The Shapley–Folkman lemma is a result in convex geometry with applications in mathematical economics that describes the Minkowski addition of sets in a vector space. Minkowski addition is defined as the addition of the sets' members: for example, adding the set consisting of the integers zero and one to itself yields the set consisting of zero, one, and two: {0, 1} + {0, 1} = {0 + 0, 0 + 1, 1 + 0, 1 + 1} = {0, 1, 2}.The Shapley–Folkman lemma and related results provide an affirmative answer to the question, "Is the sum of many sets close to being convex?" A set is defined to be convex if every line segment joining two of its points is a subset in the set: For example, the solid disk is a convex set but the circle is not, because the line segment joining two distinct points is not a subset of the circle. The Shapley–Folkman lemma suggests that if the number of summed sets exceeds the dimension of the vector space, then their Minkowski sum is approximately convex.The Shapley–Folkman lemma was introduced as a step in the proof of the Shapley–Folkman theorem, which states an upper bound on the distance between the Minkowski sum and its convex hull. The convex hull of a set Q is the smallest convex set that contains Q. This distance is zero if and only if the sum is convex. The theorem's bound on the distance depends on the dimension D and on the shapes of the summand-sets, but not on the number of summand-sets N, when N > D. The shapes of a subcollection of only D summand-sets determine the bound on the distance between the Minkowski average of N sets 1⁄N (Q1 + Q2 + ... + QN)and its convex hull. As N increases to infinity, the bound decreases to zero (for summand-sets of uniformly bounded size). The Shapley–Folkman theorem's upper bound was decreased by Starr's corollary (alternatively, the Shapley–Folkman–Starr theorem).The lemma of Lloyd Shapley and Jon Folkman was first published by the economist Ross M. Starr, who was investigating the existence of economic equilibria while studying with Kenneth Arrow. In his paper, Starr studied a convexified economy, in which non-convex sets were replaced by their convex hulls; Starr proved that the convexified economy has equilibria that are closely approximated by "quasi-equilibria" of the original economy; moreover, he proved that every quasi-equilbrium has many of the optimal properties of true equilibria, which are proved to exist for convex economies. Following Starr's 1969 paper, the Shapley–Folkman–Starr results have been widely used to show that central results of (convex) economic theory are good approximations to large economies with non-convexities; for example, quasi-equilibria closely approximate equilibria of a convexified economy. "The derivation of these results in general form has been one of the major achievements of postwar economic theory", wrote Roger Guesnerie. The topic of non-convex sets in economics has been studied by many Nobel laureates, besides Lloyd Shapley who won the prize in 2012: Arrow (1972), Robert Aumann (2005), Gérard Debreu (1983), Tjalling Koopmans (1975), Paul Krugman (2008), and Paul Samuelson (1970); the complementary topic of convex sets in economics has been emphasized by these laureates, along with Leonid Hurwicz, Leonid Kantorovich (1975), and Robert Solow (1987).The Shapley–Folkman lemma has applications also in optimization and probability theory. In optimization theory, the Shapley–Folkman lemma has been used to explain the successful solution of minimization problems that are sums of many functions. The Shapley–Folkman lemma has also been used in proofs of the "law of averages" for random sets, a theorem that had been proved for only convex sets.".
• Shapley–Folkman_lemma thumbnail Shapley–Folkman_lemma.svg?width=300.
• Shapley–Folkman_lemma wikiPageID "29237460".
• Shapley–Folkman_lemma wikiPageRevisionID "597228516".
• Shapley–Folkman_lemma alt "Illustration of a convex set, which looks somewhat like a disk: A convex set contains the line-segment joining the points x and y. The entire line-segment is a subset of the convex set.".
• Shapley–Folkman_lemma alt "Illustration of a green non-convex set, which looks somewhat like a boomerang or cashew nut. The black line-segment joins the points x and y of the green non-convex set. Part of the line segment is not contained in the green non-convex set.".
• Shapley–Folkman_lemma caption "In a convex set Q, the line segment connecting any two of its points is a subset of Q.".
• Shapley–Folkman_lemma caption "In a non-convex set Q, a point in some line-segment joining two of its points is not a member of Q.".
• Shapley–Folkman_lemma footer "Line segments test whether a subset be convex.".
• Shapley–Folkman_lemma image "Convex polygon illustration1.png".
• Shapley–Folkman_lemma image "Convex polygon illustration2.png".
• Shapley–Folkman_lemma width "155".
• Shapley–Folkman_lemma subject Category:Additive_combinatorics.
• Shapley–Folkman_lemma subject Category:Convex_geometry.
• Shapley–Folkman_lemma subject Category:Convex_hulls.
• Shapley–Folkman_lemma subject Category:Convexity_in_economics.
• Shapley–Folkman_lemma subject Category:General_equilibrium_and_disequilibrium.
• Shapley–Folkman_lemma subject Category:Geometric_transversal_theory.
• Shapley–Folkman_lemma subject Category:Mathematical_and_quantitative_methods_(economics).
• Shapley–Folkman_lemma subject Category:Mathematical_economics.
• Shapley–Folkman_lemma subject Category:Sumsets.
• Shapley–Folkman_lemma subject Category:Theorems_in_geometry.
• Shapley–Folkman_lemma comment "The Shapley–Folkman lemma is a result in convex geometry with applications in mathematical economics that describes the Minkowski addition of sets in a vector space.".
• Shapley–Folkman_lemma label "Shapley–Folkman lemma".
• Shapley–Folkman_lemma label "Лемма Шепли — Фолкмана".
• Shapley–Folkman_lemma sameAs Shapley%E2%80%93Folkman_lemma.
• Shapley–Folkman_lemma sameAs Λήμμα_των_Σάπλεϊ-Φόλκμαν.
• Shapley–Folkman_lemma sameAs Q3788045.
• Shapley–Folkman_lemma sameAs Q3788045.
• Shapley–Folkman_lemma wasDerivedFrom Shapley–Folkman_lemma?oldid=597228516.
• Shapley–Folkman_lemma depiction Shapley–Folkman_lemma.svg.