On The Heine-Borel Property and Minimum Enclosing Balls

TL;DR

This paper proves that minimum radius balls in certain metric spaces are LP type, extending the property to weak and non-symmetric spaces, and provides methods for computing these balls in specific metrics.

Contribution

It introduces a proof that minimum radius balls are LP type in various metric spaces, including weak and non-symmetric ones, and offers explicit computation primitives.

Findings

Minimum radius balls are LP type in Heine-Borel spaces.

Weak metric spaces with fixed direction also have LP type property.

Topology of the Thompson metric coincides with the Hilbert metric.

Abstract

In this paper, we contribute a proof that minimum radius balls over metric spaces with the Heine-Borel property are always LP type. Additionally, we prove that weak metric spaces, those without symmetry, also have this property if we fix the direction in which we take their distances from the centers of the balls. We use this to prove that the minimum radius ball problem is LP type in the Hilbert and Thompson metrics and Funk weak metric. In doing so, we contribute a proof that the topology induced by the Thompson metric coincides with the Hilbert. We provide explicit primitives for computing the minimum radius ball in the Hilbert metric.