A simple proof of the Shapley-Folkman theorem
\textit{R. M. Starr} [Econometrica 37, 25-38 (1969; Zbl 0172.446)] first introduced into the literature the Shapley-Folkman theorem, which later proves to be extremely important in dealing with non-convexity in economic models of large but finite agents. Most published of the Shapley-Folkman theorem use the Krein-Milman theorem or the Minkowski theorem. In this note I provide a simple proof that uses the following fact from linear algebra: Assume that \(u,u^ 1,\dots, u^ p\) are vectors in \(\mathbb{R}^ q\). If \(u\) is a nonnegative combination of \(\{u^ 1,\dots, u^ p\}\), then it must be a nonnegative combination of no more than \(q\) vectors from \(\{u^ 1,\dots, u^ p\}\). The proof of this fact can be found in any book on linear programming. This fact has been used in a standard proof of the Caratheodory Theorem. I now show that it can also be used to prove the Shapley-Folkman theorem.
scientific article; zbMATH DE number 695676