Shapley–Folkman Theorem (original) (raw)

Abstract

The Shapley–Folkman theorem places an upper bound on the size of the non-convexities (loosely speaking, openings or holes) in a sum of non-convex sets in Euclidean _N_-dimensional space, R N. The bound is based on the size of nonconvexities in the sets summed and the dimension of the space. When the number of sets in the sum is large, the bound is independent of the number of sets summed, depending rather on N, the dimension of the space. Hence the size of the non-convexity in the sum becomes small as a proportion of the number of sets summed; the non-convexity per summand goes to zero as the number of summands becomes large. The Shapley–Folkman theorem can be viewed as a discrete counterpart to the Lyapunov theorem on non-atomic measures (Grodal [2002](#ref-CR11 "Grodal, B. 2002. The equivalence principle. In Optimization and Operation Research, Encyclopedia of Life Support Systems (EOLSS), ed. U. Derigs. Cologne. Online. Available at http://www.econ.ku.dk/grodal/EOLSS-final.pdf

        . Accessed 5 Apr 2007.")).

This chapter was originally published in The New Palgrave Dictionary of Economics, 2nd edition, 2008. Edited by Steven N. Durlauf and Lawrence E. Blume

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Starr, R.M. (2008). Shapley–Folkman Theorem. In: The New Palgrave Dictionary of Economics. Palgrave Macmillan, London. https://doi.org/10.1057/978-1-349-95121-5\_1904-2

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• DOI: https://doi.org/10.1057/978-1-349-95121-5\_1904-2
• Received: 13 January 2017
• Accepted: 13 January 2017
• Published: 22 March 2017
• Publisher Name: Palgrave Macmillan, London
• Online ISBN: 978-1-349-95121-5
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Shapley–Folkman Theorem
Published:
22 March 2017
DOI: https://doi.org/10.1057/978-1-349-95121-5\_1904-2 2. #### Original
Shapley–Folkman Theorem
Published:
31 October 2016
DOI: https://doi.org/10.1057/978-1-349-95121-5\_1904-1