Calculus of Variations and Geometric Measure Theory

M. Basok - D. Cherkashin - N. Rastegaev - Y. Teplitskaya

On uniqueness in Steiner problem

created by teplitskaya1 on 10 Feb 2023
modified by cherkashin on 04 Mar 2024


Published Paper

Inserted: 10 feb 2023
Last Updated: 4 mar 2024

Journal: International Mathematics Research Notices
Pages: rnae025
Year: 2024
Doi: 10.1093/imrn/rnae025

ArXiv: 1809.01463 PDF


We prove that the set of $n$-point configurations for which solution of the planar Steiner problem is not unique has Hausdorff dimension is at most $2n-1$. Moreover, we show that the Hausdorff dimension of $n$-points configurations on which some locally minimal trees have the same length is also at most $2n-1$. Methods we use essentially requires some analytic structure and some finiteness, so that we prove a similar result for a complete Riemannian analytic manifolds under some apriori assumption on the Steiner problem on them.

Keywords: uniqueness, Steiner problem, Steiner tree, universality