Steiner Ratio and Steiner-Gromov Ratio of Gromov-Hausdorff Space

TL;DR

This paper studies the Gromov-Hausdorff space of compact metric spaces, proving that certain Steiner minimal trees are minimal fillings and establishing that both the Steiner ratio and Gromov-Steiner ratio are 1/2.

Contribution

It demonstrates that in the Gromov-Hausdorff space, Steiner minimal trees are minimal fillings under specific conditions and calculates key ratios as 1/2.

Findings

Steiner ratio of the space is 1/2

Gromov-Steiner ratio of the space is 1/2

Steiner minimal trees are minimal fillings near generic spaces

Abstract

In the present paper we investigate the metric space $\cal M$ consisting of isometry classes of compact metric spaces, endowed with the Gromov-Hausdorff metric. We show that for any finite subset $M$ from a sufficiently small neighborhood of a generic finite metric space, providing $M$ consists of finite metric spaces with the same number of points, each Steiner minimal tree in $\cal M$ connecting $M$ is a minimal filling for $M$. As a consequence, we prove that the both Steiner ratio and Gromov-Steiner ratio of $\cal M$ are equal to $1/2$.