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arXiv · 2506.18534

Compression Maps Between Polytopes

José Ayala·David Kirszenblat·J. Hyam Rubinstein
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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.

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José Ayala, David Kirszenblat, J. Hyam Rubinstein. 2025-06-23. Compression Maps Between Polytopes. https://arxiv.org/abs/2506.18534

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