On the Duality of Coverings in Hilbert Geometry

TL;DR

This paper establishes a duality principle for covering problems in Hilbert geometry, linking the minimal number of Hilbert balls needed to cover convex bodies and their polars, extending classical volumetric dualities.

Contribution

It proves a Hilbert-geometric analogue of the K"{o}nig-Milman covering duality, including new boundary-covering duality results and an alternative proof of Faifman's bounds.

Findings

Established a duality relation for Hilbert ball coverings of convex bodies.

Recovered classical volumetric duality for translative coverings.

Provided an alternative proof of Faifman's bounds in Funk and Hilbert geometries.

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 $N^H_K(G,\alpha)$ and $S^H_K(G,\alpha)$ denote, respectively, the minimum numbers of radius-$\alpha$ 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 K\"{o}nig-Milman covering duality: there exists an absolute constant $c \geq 1$ such that for any $\alpha \in (0,1]$, \[ c^{-d}\,N^H_{G^{\circ}}(K^{\circ},\alpha) ~ \leq ~ N^H_K(G,\alpha) ~ \leq ~ c^{d}\,N^H_{G^{\circ}}(K^{\circ},\alpha), \] and likewise, \[ c^{-d}\,S^H_{G^{\circ}}(K^{\circ},\alpha) ~ \leq ~ S^H_K(G,\alpha) ~ \leq ~ c^{d}\,S^H_{G^{\circ}}(K^{\circ},\alpha). \] 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 $\alpha$-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.