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arXiv · 1507.05114

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

György Pál Gehér
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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 $d$.

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BibTeXRIS

György Pál Gehér. 2015-07-15. Is it possible to determine a point lying in a simplex if we know the distances from the vertices?. https://doi.org/10.1016/j.jmaa.2016.03.024

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