Hollow polytopes with many vertices

TL;DR

The paper studies hollow polytopes within lattice settings, focusing on how many vertices such polytopes can have under natural restrictions. It develops two main directions: vertex bounds for hollow polytopes with simplicial facets or with vertices in general position, and bounds for polytopes that avoid long lattice segments; both use flatness-based width arguments and connections to Doignon's theorem. The authors define and bound hol_gp($\mathbb{Z}^d$) and hol_triangle($\mathbb{Z}^d$), showing finiteness and giving asymptotic upper bounds $O(d^2(\log d)^3)$ for the general-position case and a universal $2^d$ bound in the simplicial case, as well as constructions achieving near-optimal lower bounds. They conclude with open questions about tightening asymptotics and extending to broader configurations.

Abstract

Given a set $S \subseteq \mathbb{R}^d$, a hollow polytope has vertices in $S$ but contains no other point of $S$ in its interior. We prove upper and lower bounds on the maximum number of vertices of hollow polytopes whose facets are simplices or whose vertices are in general position. We also obtain relatively tight asymptotic bounds for polytopes which do not contain lattice segments of large length.

Theorems & Definitions (20)

• Theorem : Hoffman Hoffman1979
• Theorem
• Theorem 1.1
• Theorem 1.2
• proof
• Theorem 2.2
• proof
• Definition 2.3
• Theorem 2.4: Flatness theorem, Reis--Rothvoss flatness-thm
• Theorem 2.5