Five-dimensional Perfect Simplices

TL;DR

This paper establishes the existence of five-dimensional perfect simplices inscribed in the unit cube, determines their minimal homothety ratios, and extends the understanding of such simplices in higher dimensions.

Contribution

It proves the existence of perfect simplices in five and nine dimensions and describes infinite families of extremal simplices for certain dimensions, advancing geometric understanding of simplices within cubes.

Findings

Existence of perfect simplices in ${\mathbb R}^5$ and ${\mathbb R}^9$.

Exact values of $\xi_n$ for $n=5$ and $n=9$.

Infinite families of extremal simplices for $n=5,7,9$.

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