Equiprojective polytopes in higher dimension

TL;DR

This paper extends the concept of equiprojective polytopes to higher dimensions, providing bounds on their types and exploring their topological properties, with implications for algorithms like Shadow Vertex.

Contribution

It generalizes equiprojectivity to higher dimensions, establishes bounds on combinatorial types, and proves pathwise connectedness in certain Grassmannian subsets.

Findings

Lower bounds on the number of equiprojective polytope types

Pathwise connectedness of a subset of the Grassmannian

Implications for the complexity of the Shadow Vertex algorithm

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.