Representability of convex sets by analytical linear inequality systems

Abstract

The solution sets of analytical linear inequality systems posed in the Euclidean space form a transition class between the polyhedral convex sets and the closed convex sets, which are representable by means of linear continuous systems. The constraint systems of many semi-infinite programming problems are analytical, and their feasible sets retain geometric properties of the polyhedral sets which are useful in the numerical treatment of such kind of optimization problems. The Euclidean closed n-dimensional balls admit analytical representation if and only if n<3. This paper solves, in a negative way, the analytical representation problem for a wide class of n-dimensional convex sets, with n⩾3, which includes quasi-polyhedral sets and smooth convex bodies.