Fermat-Steiner Problem in the Metric Space of Compact Sets endowed with Hausdorff Distance

TL;DR

This paper investigates the Fermat-Steiner problem within the space of compact sets equipped with the Hausdorff distance, providing descriptions of minimal configurations and Steiner trees, including examples with unusual symmetry properties.

Contribution

It offers a description of all compact sets minimizing the Fermat-Steiner problem in the Hausdorff space and characterizes Steiner minimal trees for three-point boundaries.

Findings

Characterization of minimizers in the Hausdorff space

Description of Steiner minimal trees for three points

Example of a triangle with asymmetric shortest trees

Abstract

The Fermat-Steiner problem consists in finding all points in a metric space $Y$ such that the sum of distances from each of them to the points from some fixed finite subset of $Y$ is minimal. This problem is investigated for the metric space $Y=H(X)$ of compact subsets of a metric space $X$, endowed with the Hausdorff distance. For the case of a proper metric space $X$ a description of all compacts $K\in H(X)$ which the minimum is attained at is obtained. In particular, the Steiner minimal trees for three-element boundaries are described. We also construct an example of a regular triangle in $H(R^2)$, such that all its shortest trees have no "natural" symmetry.