Is it possible to determine a point lying in a simplex if we know the distances from the vertices?

TL;DR

This paper explores whether distances from vertices of a simplex uniquely determine a point within it in various normed spaces, extending a known Euclidean fact to more general settings.

Contribution

It characterizes the normed spaces where distances to simplex vertices uniquely identify points inside the simplex, revealing a dependence on the dimension.

Findings

Characterization depends on the dimension d

Distances determine points uniquely in certain normed spaces

Results extend Euclidean distance properties to general normed spaces

Abstract

It is an elementary fact that if we fix an arbitrary set of $d+1$ affine independent points $\{p_0,\dots p_d\}$ in $\mathbb{R}^d$, then the Euclidean distances $\{|x-p_j|\}_{j=0}^d$ determine the point $x$ in $\mathbb{R}^d$ uniquely. In this paper we investigate a similar problem in general normed spaces which is motivated by this known fact. Namely, we characterize those, at least $d$-dimensional, real normed spaces $(X, \|\cdot\|)$ such that for every set of $d+1$ affine independent points $\{p_0,\dots p_d\} \subset X$, the distances $\{\|x-p_j\|\}_{j=0}^d$ determines the point $x$ lying in the simplex $\mathrm{Conv}(p_0,\dots p_d)$ uniquely. Surprisingly, the characterization depends on $d$.