Helly-type decomposition theorems for convex sets (Q1088158)

Given two subsets A and B of \({\mathbb{R}}^ d\) (d\(\geq 2)\), B is called a summand of A if and only if \(A= B+C:= \{x+y| x\in B, y\in C\}\) for some subset C of \({\mathbb{R}}^ d\). In this paper, Helly's theorem is used to characterize the convex summands of a convex set and to prove Helly- type theorems concerning the stability of the summand relation. The method based on intersection properties enables one to drop boundedness restrictions and in a few cases also closedness assumptions.