## Shapley–Folkman theorem

From The New Palgrave Dictionary of Economics, Second Edition, 2008
Edited by
Alternate versions available: 1987 Edition

### 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, RN. The bound is based on the size of non-convexities 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).