Reconstruction of norm by geometry of minimal networks

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