Strong CHIP for infinite systems of convex sets in normed linear spaces

Abstract

This article studies the strong conical hull intersection property (strong CHIP) for a general (possibly infinite) system of convex sets in normed linear spaces. The first part of the article presents necessary and sufficient conditions for strong CHIP in terms of tangent cones and feasible direction cones in the primal space, and reveals some differences between strong CHIP for finite and infinite systems. The second part of the paper proves that strong CHIP has the segment extension property, i.e. if strong CHIP is satisfied at some ‘base points’ then it can be extended from the base points along line segments to the entire set. Thus, the verification effort of strong CHIP can be greatly reduced in many cases. For example, if the intersection set is a translated cone, then strong CHIP at the vertex ensures strong CHIP for the entire cone. In the special case of Euclidean space, strong CHIP at the extreme points of the intersection set ensures strong CHIP for the entire intersection.