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 + 1 affine independent points { p 0 , … , p d } in R d , then the Euclidean distances { | x − p j | } j = 0 d determine the point x in R d uniquely. In this paper we investigate a similar problem in general normed spaces which is motivate...
Ausführliche Beschreibung

It is an elementary fact that if we fix an arbitrary set of d + 1 affine independent points { p 0 , … , p d } in R d , then the Euclidean distances { | x − p j | } j = 0 d determine the point x in 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 , ‖ ⋅ ‖ ) for which every set of d + 1 affine independent points { p 0 , … , p d } ⊂ X , the distances { ‖ x − p j ‖ } j = 0 d determine the point x lying in the simplex Conv ( { p 0 , … , p d } ) uniquely. If d = 2 , then this condition is equivalent to strict convexity, but if d > 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 () is re-proven using the fundamental theorem of projective geometry. Ausführliche Beschreibung