Extend exercise 3.210 to allow for noncompact Si, thus proving the Shapley-Folkman theorem.
Exercise 3.210
Let {S1, S2, . . . , Sn} be a collection of nonempty compact subsets of an m-dimensional linear space, and let x ˆˆ conv

We consider the Cartesian product of the convex hulls of Si, namely

Every point in P is an n-tuple (x1, x2, . . . , xn) where each xi belongs to the corresponding conv Si. Let P(x) denote the subset of P for which

P - IIconv S, 2l Xi = X, that is, P(x) = (Xi, X2, . . . , Xn ) : Xi E conv S, and x,-x