# On The Heine-Borel Property and Minimum Enclosing Balls

arXiv: https://arxiv.org/abs/2412.17138 · HTML: https://mlsanity.com/paper/2412.17138/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Dec 2024.

## Built on by (most cited first, 1 of 1)

- [Hilbert geometry of the symmetric positive-definite bicone: Application to the geometry of the extended Gaussian family](https://mlsanity.com/paper/2508.14369/) (2025)

## Abstract (arXiv)

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.