Equiprojective polytopes in higher dimension

TL;DR

The work extends equiprojectivity from 3D to $d$-dimensional polytopes by defining $k$-equiprojectivity via projections to admissible planes and by formalising degeneracy concepts tied to 2-faces. It establishes a lower bound on the number of combinatorial types for even $k$-equiprojective polytopes through zonotopes and Buffière–Pournin results, and develops a pathwise, controlled-connectivity framework in the Grassmannian to extend the Hasan–Lubiw characterisation to higher dimensions. Central to the theory is the edge-2-face compensation criterion, proving that a polytope is equiprojective iff its edge-2-faces can be partitioned into compensating pairs, with constructive lemmas that handle degeneracies and visibility chains. The constructions showing many simultaneous degeneracies and the resulting implications for projection complexity offer insights relevant to the Shadow Vertex algorithm and broaden the reverse-projection viewpoint on polytope projections.

Abstract

A 3-dimensional polytope is called k-equiprojective if every planar projection along a direction non-parallel to any facet is a k-gon. In this article, we generalise equiprojectivity to higher dimensions and give a lower bound on the number of combinatorial types of equiprojective polytopes. We also establish the pathwise connectedness of a subset of the Grassmannian in the case of (d-2)-dimensional spaces with conditions on the explicit path. This makes it possible to extend the Hasan--Lubiw characterisation of equiprojectivity to higher dimensions. Equiprojectivity provides cases relevant to the study of the Shadow Vertex algorithm, showing there is no hope minimising the complexity of the projection. It also offers a reverse point of view on the usual study of planar projections of polytopes as the projections have a fixed size.

Figures (9)

• Figure 1: Example of degenerated projection of the cube with three vertices that are sent on aligned points on the boundary
• Figure 2: A degenerated projection of the hypercube
• Figure 3: A perturbed hypercube providing a counter-example
• Figure 5: In blue, different examples of projection of $Y_\theta$
• Figure 6: Example of the projection of $P_4$ on $W$. In red, the degenerating 2-faces.

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2
• Definition 1
• Remark
• Definition 2: Admissibility
• Remark
• Remark
• proof : Proof of observation and bound \ref{['borne']}
• Definition 3
• Remark