High Quality Content by WIKIPEDIA articles! In convex geometry and mathematical economics, the Shapley–Folkman lemma and the closely related Shapley–Folkman–Starr theorem state that the Minkowski sum of many non-convex subsets of a finite-dimensional vector space is nearly convex. The Shapley–Folkman results are named after Lloyd Shapley and Jon Folkman, who proved both the Shapley–Folkman lemma and a weaker version of the Shapley–Folkman–Starr theorem in an unpublished report, "Starr's problem" (1966), which was cited by Starr. As Starr showed, these results may be used to show the existence of approximate equilibria for nonconvex economies. His research on this problem began while he was an undergraduate at Stanford University, where Starr had enrolled in the (graduate) advanced mathematical economics course of Kenneth J. Arrow.

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