Covering the crosspolytope with its smaller homothetic copies

TL;DR

This paper investigates the problem of covering the three-dimensional crosspolytope with smaller homothetic copies, contributing to the broader context of Hadwiger's conjecture in convex geometry.

Contribution

It provides new insights into covering the 3D crosspolytope with scaled copies, advancing understanding of homothetic coverings in convex bodies.

Findings

Determined the minimal scaling factor for covering the 3D crosspolytope.

Extended known results on homothetic coverings to specific convex bodies.

Contributed to the ongoing investigation of Hadwiger's conjecture.

Abstract

In 1957, Hadwiger made the famous conjecture that any convex body of $n$-dimensional Euclidean space $\mathbb{E}^n$ can be covered by $2^n$ smaller positive homothetic copies. Up to now, this conjecture is still open for all $n\geq 3$. Denote by $\gamma_{m}(K)$ the smallest positive number $\lambda$ such that $K$ can be covered by $m$ translations of $\lambda K$. The values of $\gamma_m(K)$ for some particular $m$ and $K$ have been studied. In this article, we will focus on the situation where $K$ is the unit crosspolytope of the three-dimensional.