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arXiv · 2103.10004

Covering the crosspolytope with its smaller homothetic copies

Yanlu Lian·Yuqin zhang
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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.

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Yanlu Lian, Yuqin zhang. 2021-03-18. Covering the crosspolytope with its smaller homothetic copies. https://arxiv.org/abs/2103.10004

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