Compression Maps Between Polytopes

TL;DR

This paper characterizes when a piecewise linear map between polytope shapes is a compression or weak compression, introduces a partial order on shapes, and constructs a compression metric with completeness results for simplexes.

Contribution

It provides necessary and sufficient conditions for compression maps between polytope shapes and develops a new metric on the shape space, including completeness properties.

Findings

Characterization of compression and weak compression maps.

Definition of a partial order on polytope shapes.

Construction of a compression metric with completeness for simplexes.

Abstract

A shape of a combinatorial polytope is a convex embedding into Euclidean space. We provide necessary and sufficient conditions for a piecewise linear map between two shapes of the same polytope to be a compression (respectively a weak compression), meaning a distance decreasing (respectively a distance non-increasing) map between distinct pairs of points. We establish that there is a partial order on the space of shapes given by the relation of having a weak compression map between pairs of shapes. Finally, we construct a compression metric on the projective shape space of a polytope; the space of convex Euclidean realizations modulo rigid motions and homothety. For the projective space shape of a simplex, we show that the compression metric is complete. For general polytopes, we establish that the projective shape space has a natural completion given by the projective shape space of weakly convex realizations.