Five-dimensional Perfect Simplices

Abstract

Let $Q_n=[0,1{]}^n$ be the unit cube in ${\mathbb{R}}^n$, $n\in \mathbb{N}$. For a nondegenerate simplex $S\subset {\mathbb{R}}^n$, consider the value $\xi (S)=\min \{\sigma >0:Q_n\subset \sigma S\}$. Here $\sigma S$ is a homothetic image of $S$ with homothety center at the center of gravity of $S$ and coefficient of homothety $\sigma$. Let us introduce the value ${\xi }_n=\min \{\xi (S):S\subset Q_n\}$. We call $S$ a perfect simplex if $S\subset Q_n$ and $Q_n$ is inscribed into the simplex ${\xi }_{n}S$. It is known that such simplices exist for $n=1$ and $n=3$. The exact values of ${\xi }_n$ are known for $n=2$ and in the case when there exist an Hadamard matrix of order $n+1$, in the latter situation ${\xi }_n=n$. In this paper we show that ${\xi }_5=5$ and ${\xi }_9=9$. We also describe infinite families of simplices $S\subset Q_n$ such that $\xi (S)={\xi }_n$ for $n=5,7,9$. The main result of the paper is the existence of perfect simplices in ${\mathbb{R}}^5$. Keywords: simplex, cube, homothety, axial diameter, Hadamard matrix

25 pages, 4 figures

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