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HV-symmetric polyhedra and bipolarity

David Avis

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 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.

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 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.
Paper Structure (1 section, 3 theorems, 13 equations)

This paper contains 1 section, 3 theorems, 13 equations.

Table of Contents

  1. Acknowledgment

Key Result

Lemma 1

Let $P$ be a pointed polyhedron with $\mathcal{V}(P)$ encoded as $V(P)$ (vr). Then and so $V(P)$ is an encoding of $\mathcal{H}(P^+)$.

Theorems & Definitions (9)

  • Definition 1
  • Example 1
  • Example 2
  • Lemma 1
  • proof
  • Lemma 2: Extended bipolar equation
  • proof
  • Theorem 1
  • proof