Distances from the vertices of a regular simplex

TL;DR

This paper proves a unique relation among distances from any point to the vertices of a regular simplex in Euclidean space, using analysis, algebra, and geometry to establish its exclusivity.

Contribution

It demonstrates that the given distance relation is the only possible relation among these distances for a regular simplex.

Findings

The relation holds universally for any point in Euclidean space.

The relation is unique among all possible relations.

The proof combines tools from analysis, algebra, and geometry.

Abstract

If $S$ is a given regular $d$-simplex of edge length $a$ in the $d$-dimensional Euclidean space $\mathcal{E}$, then the distances $t_1$, $\cdots$, $t_{d+1}$ of an arbitrary point in $\mathcal{E}$ to the vertices of $S$ are related by the elegant relation $$(d+1)\left( a^4+t_1^4+\cdots+t_{d+1}^4\right)=\left( a^2+t_1^2+\cdots+t_{d+1}^2\right)^2.$$ The purpose of this paper is to prove that this is essentially the only relation that exists among $t_1,\cdots,t_{d+1}.$ The proof uses tools from analysis, algebra, and geometry.