For a convex set S, we study the facial structure of its integer hull, S_Z. Crucial to our study is the decomposition of the integer hull into the convex hull of its extreme points, conv(ext(S_Z)), and its recession cone. Although conv(ext(S_Z)) 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(S_Z)) has an infinite number of facets. Using these results, we provide a necessary and sufficient condition for semidefinite representability of conv(ext(S_Z)).
展开▼
