Shapley–Folkman lemma

The Shapley–Folkman lemma is a result in convex geometry that describes the Minkowski addition of sets in a vector space. It is named after mathematicians Lloyd Shapley and Jon Folkman, but was first published by the economist Ross M. Starr.

The lemma may be intuitively understood as saying that, if the number of summed sets exceeds the dimension of the vector space, then their Minkowski sum is approximately convex.

Related results provide more refined statements about how close the approximation is. For example, the Shapley–Folkman theorem provides an upper bound on the distance between any point in the Minkowski sum and its convex hull. This upper bound is sharpened by the Shapley–Folkman–Starr theorem (alternatively, Starr's corollary).

The Shapley–Folkman lemma has applications in economics, optimization and probability theory. In economics, it can be used to extend results proved for convex preferences to non-convex preferences. In optimization theory, it can be used to explain the successful solution of minimization problems that are sums of many functions. In probability, it can be used to prove a law of large numbers for random sets.