HV-symmetric polyhedra and bipolarity

Abstract

A polyhedron is pointed if it contains at least one vertex. Every pointed polyhedron $P$ in $R^n$ can be described by an $H-representation$ $H\left(P\right)$ consisting of half spaces or equivalently by a $V$-representation $V\left(P\right)$ consisting of the convex hull of a set of vertices and extreme rays. We can define matrices $H\left(P\right)$ and $V\left(P\right)$, each with $n+1$ columns, that encode these representations. Define polyhedron $Q$ by setting $H\left(Q\right)=V\left(P\right)$. We show that $Q$ is the polar of $P$. Call $P$ $HV$-symmetric if $Q$ is pointed and $V\left(Q\right)$ in turn encodes $H\left(P\right)$. 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 prove this using Minkowski’s bipolar equation and discuss implications for the vertex and facet enumeration problems.