On the Duality of Coverings in Hilbert Geometry

2026-03-19

AI Summary

This study investigates the minimal covering problem for convex bodies and their boundaries in Hilbert geometry, establishing—for the first time—a polarity-based duality theory for covering numbers under the non-translation-invariant Hilbert metric. By introducing key tools including α-dilations, stability lemmas for polarity and dilations, a local relative isoperimetric inequality, and estimates of Holmes–Thompson area, the authors demonstrate that the covering numbers of a convex body and its polar dual mutually control each other up to exponential factors, with bounds governed by absolute constants. The main contributions include a novel duality relation for boundary coverings, an extension of classical volume duality to the setting of Hilbert geometry, and a new proof of Faifman’s polarity bound.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization

Abstract

We prove polarity duality for covering problems in Hilbert geometry. Let $G$ and $K$ be convex bodies in $\mathbb{R}^d$ where $G \subset \operatorname{int}(K)$ and $\operatorname{int}(G)$ contains the origin. Let $N^H_K(G,α)$ and $S^H_K(G,α)$ denote, respectively, the minimum numbers of radius-$α$ Hilbert balls in the geometry induced by $K$ needed to cover $G$ and $\partial G$. Our main result is a Hilbert-geometric analogue of the König-Milman covering duality: there exists an absolute constant $c \geq 1$ such that for any $α\in (0,1]$, \[ c^{-d}\,N^H_{G^{\circ}}(K^{\circ},α) ~ \leq ~ N^H_K(G,α) ~ \leq ~ c^{d}\,N^H_{G^{\circ}}(K^{\circ},α), \] and likewise, \[ c^{-d}\,S^H_{G^{\circ}}(K^{\circ},α) ~ \leq ~ S^H_K(G,α) ~ \leq ~ c^{d}\,S^H_{G^{\circ}}(K^{\circ},α). \] We also recover the classical volumetric duality for translative coverings of centered convex bodies, and obtain a new boundary-covering duality in that setting. The Hilbert setting is subtler than the translative one because the metric is not translation invariant, and the local Finsler unit ball depends on the base point. The proof involves several ideas, including $α$-expansions, a stability lemma that controls the interaction between polarity and expansion, and, in the boundary case, a localized relative isoperimetric argument combined with Holmes--Thompson area estimates. In addition, we provide an alternative proof of Faifman's polarity bounds for Holmes--Thompson volume and area in the Funk and Hilbert geometries.

Problem

• Hilbert geometry
• covering duality
• convex bodies
• polarity
• Finsler metric

Innovation

• Hilbert geometry
• covering duality
• polarity
• Holmes–Thompson volume
• α-expansion