On Uniqueness in Steiner Problem

Abstract

Abstract We prove that the set of $n$-point configurations for which the solution to the planar Steiner problem is not unique has the Hausdorff dimension at most $2n-1$ (as a subset of $\mathbb{R}^{2n}$). Moreover, we show that the Hausdorff dimension of the set of $n$-point configurations for which at least two locally minimal trees have the same length is also at most $2n-1$. The methods we use essentially rely upon the theory of subanalytic sets developed in [ 1]. Motivated by this approach, we develop a general setup for the similar problem of uniqueness of the Steiner tree where the Euclidean plane is replaced by an arbitrary analytic Riemannian manifold $M$. In this setup, we argue that the set of configurations possessing two locally-minimal trees of the same length either has dimension equal to $n \dim M - 1$ or has a non-empty interior. We provide an example of a two-dimensional surface for which the last alternative holds. In addition to the above-mentioned results, we study the set of $n$-point configurations for which there is a unique solution to the Steiner problem in $\mathbb{R}^{d}$. We show that this set is path-connected.