Five-dimensional Perfect Simplices

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