The width of 5-dimensional prismatoids

TL;DR

This paper investigates the width of 5-dimensional prismatoids, demonstrating the existence of unbounded width and constructing smaller examples that lead to non-Hirsch polytopes in lower dimensions.

Contribution

It provides the first constructions of 5-prismatoids with large width and fewer vertices, showing the unbounded nature of their width.

Findings

Existence of 5-prismatoids with width six and only 25 vertices.

Construction of non-Hirsch polytopes in dimension 20.

Width of 5-prismatoids is unbounded, growing with the number of vertices.

Abstract

Santos' construction of counter-examples to the Hirsch Conjecture (2012) is based on the existence of prismatoids of dimension d of width greater than d. Santos, Stephen and Thomas (2012) have shown that this cannot occur in $d \le 4$. Motivated by this we here study the width of 5-dimensional prismatoids, obtaining the following results: - There are 5-prismatoids of width six with only 25 vertices, versus the 48 vertices in Santos' original construction. This leads to non-Hirsch polytopes of dimension 20, rather than the original dimension 43. - There are 5-prismatoids with $n$ vertices and width $\Omega(\sqrt{n})$ for arbitrarily large $n$. Hence, the width of 5-prismatoids is unbounded.