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

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 $(X,\Vert \cdot \Vert )$ such that for every set of $d+1$ affine independent points $\{p_0,\dots p_d\}\subset X$, the distances $\{\Vert x-p_j\Vert {\}}_{j=0}^d$ determines the point $x$ lying in the simplex $\mathrm{Conv}(p_0,\dots p_d)$ uniquely. Surprisingly, the characterization depends on $d$.

14 pages, 3 figures The first Arxiv version had a different title!

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