On Uniqueness in Steiner Problem

International Mathematics Research Notices, vol. 2024, pp. 8819–8838

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.

Authors 4

1. Mikhail Basok

   University of Helsinki

   Affiliation as printed

   University of Helsinki , 00500 Helsinki, Finland

2. Danila Dmitrievich Cherkashin

   Bulgarian Academy of Sciences · Institute of Mathematics and Informatics

   Affiliation as printed

   Chebyshev Laboratory, 199178 St.Petersburg , Russia

   Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences , 1113 Sofia, Bulgaria

3. Nikita Rastegaev

   Russian Academy of Sciences · St. Petersburg Department of Steklov Institute of Mathematics · Steklov Mathematical Institute

   Affiliation as printed

   St. Petersburg Department of V.A.Steklov Institute of Mathematics of the Russian Academy of Sciences , 191023 St. Petersburg, Russia

4. Yana Teplitskaya Aachen Mathematical Institute

   Leiden University · Centre National de la Recherche Scientifique · Université Paris-Saclay · Laboratoire de Mathématiques d'Orsay

   Affiliation as printed

   Laboratoire de Mathématiques d'Orsay , Université Paris-Saclay, CNRS, 91400 Orsay, France

   Mathematical Institute, Leiden University , 2300 RA Leiden, the Netherlands