Steiner Ratio for Manifolds
D. Cieslik, A.O. Ivanov, A.A. Tuzhilin
TL;DR
This paper investigates the Steiner ratio on Riemannian manifolds, providing estimates and exact values for specific surfaces like flat tori, Klein bottles, and projective planes of constant curvature.
Contribution
It offers new estimates for the Steiner ratio on Riemannian manifolds and computes exact ratios for certain surfaces with constant curvature.
Findings
Steiner ratio estimates on Riemannian manifolds
Exact Steiner ratios for flat tori and Klein bottles
Results for projective plane of constant positive curvature
Abstract
The Steiner ratio characterizes the greatest possible deviation of the length of a minimal spanning tree from the length of the minimal Steiner tree. In this paper, estimates of the Steiner ratio on Riemannian manifolds are obtained. As a corollary, the Steiner ratio for flat tori, flat Klein bottles, and projective plane of constant positive curvature are computed. Steiner ratio - Steiner problem - Gilbert--Pollack conjecture - surfaces of constant curvature
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