HV-symmetric polyhedra and bipolarity

A polyhedron is pointed if it contains at least one vertex. Every pointed polyhedron P in R^n can be described by an H-representation consisting of half spaces or equivalently by a V-representation consisting of the convex hull of a set of vertices and extreme rays. We can define matrices H(P) and V(P), each with n + 1 columns, that encode these representations. Define polyhedron Q by setting H(Q)=V(P). We show that Q is the polar of P. Call P HV-symmetric if V(Q) in turn encodes the H-representation of P. It is well known and often stated that polytopes that contain the origin in their interior and pointed polyhedral cones are HV-symmetric. We show here that, more generally, a pointed polyhedron with pointed polar is HV-symmetric if and only if it contains the origin. We show this using Minkowski's bipolar equation and discuss implications for the vertex and facet enumeration problems.