Reconstruction of norm by geometry of minimal networks

Abstract

An inverse problem to the Steiner minimal tree searching problem in a normed space is studied. Namely, let a normed space be given and let all Steiner minimal trees be known in this space. The problem is to describe all norms with the same Steiner minimal trees for all finite boundary sets as in the given space. The paper presents a review of known results on the problem and announces the uniqueness of the set of Steiner minimal trees for any two-dimensional space with a strongly convex and differentiable norm.