Distances from the Vertices of a Regular Simplex

Abstract

If S is a given regular d-simplex of edge length a in the d-dimensional Euclidean space $\mathcal {E}$, then the distances $t_1$, $\ldots$, $t_{d+1}$ of an arbitrary point in $\mathcal {E}$ to the vertices of S are related by the elegant relation

$$\begin{aligned} (d+1)\left( a^4+t_1^4+\cdots +t_{d+1}^4\right) =\left( a^2+t_1^2+\cdots +t_{d+1}^2\right) ^2. \end{aligned}$$

The purpose of this paper is to prove that this is essentially the only relation that exists among $t_1,\ldots ,t_{d+1}.$ The proof uses tools from analysis, algebra, and geometry.