On reduced polytopes

TL;DR

This paper constructs a reduced polytope in three-dimensional space, answering a longstanding question about their existence in dimensions three and higher, and explores properties of such polytopes.

Contribution

It provides the first explicit example of a reduced polytope in D and investigates their properties, addressing a question posed by Lassak.

Findings

Existence of a reduced polytope in D confirmed

Properties of reduced polytopes in D analyzed

Answer to Lassak's question about higher dimensions

Abstract

A convex body $R$ in $\mathbb R^d$ is called reduced if the minimal width $\Delta(R')$ of each convex body $R'\subset R$ different from $R$ is strictly smaller than the minimal width $\Delta(R)$ of $R$. In this article we construct a reduced polytope in $\mathbb R^3$, i.e. we answer the following question posed by Lassak: do there exist reduced polytopes in $\mathbb R^d$, $d\geqslant3$? Also, we prove some properties of reduced polytopes in $\mathbb R^3$.