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On The Heine-Borel Property and Minimum Enclosing Balls

Hridhaan Banerjee, Carmen Isabel Day, Megan Hunleth, Sarah Hwang, Auguste H. Gezalyan, Olya Golovatskaia, Nithin Parepally, Lucy Wang, David M. Mount

TL;DR

The paper addresses the minimum enclosing radius ball problem in metric spaces with the Heine-Borel property, proving it is an LP-type problem. It extends to weak metrics when ball direction is fixed and demonstrates LP-type status for Hilbert, Thompson, and Funk metrics, including a result that the Thompson and Hilbert topologies coincide. It provides explicit primitives for computing Hilbert-radius balls and derives algorithmic time bounds, notably showing an $O(n\log^3 m)$-time LP-type algorithm for Hilbert radius balls. Overall, the work broadens the LP-type framework to non-Euclidean geometries and offers concrete computational tools for minimum enclosing balls in Hilbert, Thompson, and Funk metric spaces.

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.

On The Heine-Borel Property and Minimum Enclosing Balls

TL;DR

The paper addresses the minimum enclosing radius ball problem in metric spaces with the Heine-Borel property, proving it is an LP-type problem. It extends to weak metrics when ball direction is fixed and demonstrates LP-type status for Hilbert, Thompson, and Funk metrics, including a result that the Thompson and Hilbert topologies coincide. It provides explicit primitives for computing Hilbert-radius balls and derives algorithmic time bounds, notably showing an -time LP-type algorithm for Hilbert radius balls. Overall, the work broadens the LP-type framework to non-Euclidean geometries and offers concrete computational tools for minimum enclosing balls in Hilbert, Thompson, and Funk metric spaces.

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.

Paper Structure

This paper contains 7 sections, 15 theorems, 7 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Metric Spaces
  4. The Hilbert and Thompson Metrics, and the Funk Weak Metric
  5. Minimum Enclosing Radius Balls and the Heine-Borel Property
  6. Hilbert Radius Minimum Enclosing Ball
  7. Conclusion

Key Result

Lemma 6

Hilbert balls have at most $O(m)$ sides.

Figures (2)

  • Figure 1: (a) Funk ball, (b) Thompson ball (c) Hilbert ball, and (d) reverse Funk ball.
  • Figure 2: This is $\mathbb{R}^2$ with several closed balls removed. There is no Euclidean minimum radius ball through $x,y,z$.

Theorems & Definitions (24)

  • Definition 1: Metric Space
  • Definition 2: Closed Ball
  • Definition 3: Funk weak metric
  • Definition 4: Hilbert metric
  • Definition 5: Spoke
  • Lemma 6: Nielsen and Shao nielsen2017balls
  • Definition 7: Thompson metric
  • Lemma 8
  • Theorem 9
  • Corollary 10
  • ...and 14 more