Shapley–Folkman-type theorem for integrally convex sets

Abstract

The Shapley–Folkman theorem is a statement about the Minkowski sum of (non-convex) sets, expressing the closeness of the Minkowski sum to convexity in a quantitative manner. This paper establishes similar theorems for integrally convex sets, M${}^{♮}$-convex sets, and L${}^{♮}$-convex sets, which are major classes of discrete convex sets in discrete convex analysis.