Approximation of Points of the Convex Hull of a Sum of Sets by Points of the Sum: An Elementary Approach*

The Shapley-Folkman theorem and its corollaries [ 1, 2, 3, 4, 5, 6, 81 provide strong bounds on the distance between the sum of a family of nonconvex sets and the convex hull of the sum. Proofs of the theorem are nonconstructive, and require moderately advanced analysis. The proof developed below is based on elementary considerations. It provides an approximation sequentially with the successive addition of sets to the sum. The approximation is not so close as that provided by the Shapley-Folkman theorem, but for any given point of the convex hull we will find a specific point in the sum within a previously determined bound on the distance between the two. Particular virtues of the bounds associated with the Shapley-Folkman theorem are the relatively tight approximation developed and its behavior as the number of summands becomes large. The bounded distance between the sum and its convex hull depends not on the number of sets summed (denoted m) as this number becomes large, but rather on the dimensionality of the space (denoted N). Thus as the number of summands becomes large, the average discrepancy between the sum and its convex hull converges to 0 as l/m. The bounds developed below, on the contrary, vary as m”* so that the average discrepancy converges to 0 as l/m”*. The virtue of the results here is their comparative ease of proof and the sequential construction making it relatively easy to find a point of the sum nearby to any chosen point of the convex hull. For S CR”‘. S compact, we define several measures of size and nonconvexity. con S denotes the convex hull of 5’.