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

It is an elementary fact that if we fix an arbitrary set of d+1d+1 affine independent points {p0,…,pd}{p0,…,pd} in RdRd, then the Euclidean distances {|x−pj|}j=0d determine the point x   in RdRd 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,‖⋅‖)(X,‖⋅‖) for which every set of d+1d+1 affine independent points {p0,…,pd}⊂X{p0,…,pd}⊂X, the distances {‖x−pj‖}j=0d determine the point x   lying in the simplex Conv({p0,…,pd})Conv({p0,…,pd}) uniquely. If d=2d=2, then this condition is equivalent to strict convexity, but if d>2d>2, then surprisingly this holds only in inner product spaces. The core of our proof is some previously known geometric properties of bisectors. The most important of these (Theorem 1) is re-proven using the fundamental theorem of projective geometry.