On The Heine-Borel Property and Minimum Enclosing Balls

2024-12-22

arXiv.org

AI Summary

This paper investigates the computational complexity of the Minimum Bounding Sphere (MBS) problem in metric spaces, specifically asking whether the MBS problem is LP-type—i.e., polynomial-time reducible to linear programming—in Heine–Borel weak metric spaces, possibly asymmetric. Method: The authors employ tools from linear programming theory, weak metric geometry, convex analysis, and topology to analyze the MBS problem under Hilbert, Thompson, and Funk weak metrics. Contribution/Results: They establish, for the first time, that the MBS problem is LP-type under all three weak metrics; notably, LP-type structure persists even in asymmetric settings when distance direction is fixed. They further prove rigorously that the Thompson and Hilbert metrics induce identical topologies. A general LP-type characterization framework for MBS is developed, and computable explicit primitives for the Hilbert metric are provided. These results lay a new theoretical foundation for high-dimensional optimization over cones.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Web measurements

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.

Problem

• Prove minimum radius balls in Heine-Borel metric spaces are LP type.
• Show weak metric spaces with fixed direction distances are LP type.
• Establish topology equivalence between Thompson and Hilbert metrics.

Innovation

• Prove minimum radius balls are LP type
• Extend proof to weak metric spaces
• Provide primitives for Hilbert metric computation