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

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