Perfect simplices in ${\mathbb R}^5$

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 exists 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 confirmation of the existence of perfect simplices in ${\mathbb R}^5$.