Covering the crosspolytope with its smaller homothetic copies

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.