Facial structure and representation of integer hulls of convex sets

Abstract

For a convex set S, we study the facial structure of its integer hull, Sℤ. Crucial to our study is the decomposition of the integer hull into the convex hull of its extreme points, conv(ext(Sℤ)), and its recession cone. Although conv(ext(Sℤ)) might not be a polyhedron, or might not even be closed, we show that it shares several interesting properties with polyhedra: all faces are exposed, perfect, and isolated, and maximal faces are facets. We show that S ℤ has an infinite number of extreme points if and only if conv(ext(Sℤ)) has an infinite number of facets. Using these results, we provide a necessary and sufficient condition for semidefinite representability of conv(ext(S ℤ)). © 2013 Springer-Verlag.