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On the Duality of Coverings in Hilbert Geometry

Sunil Arya, David M. Mount

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.

On the Duality of Coverings in Hilbert Geometry

Abstract

We prove polarity duality for covering problems in Hilbert geometry. Let and be convex bodies in where and contains the origin. Let and denote, respectively, the minimum numbers of radius- Hilbert balls in the geometry induced by needed to cover and . Our main result is a Hilbert-geometric analogue of the König-Milman covering duality: there exists an absolute constant such that for any , and likewise, 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 -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.
Paper Structure (30 sections, 57 theorems, 231 equations, 5 figures)

This paper contains 30 sections, 57 theorems, 231 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Overview of Techniques
  3. Organization
  4. Preliminaries
  5. Measures and Smoothness
  6. Funk and Hilbert Geometries
  7. Covering numbers
  8. Holmes--Thompson Measures
  9. Duality of Holmes--Thompson Measures
  10. Localization and Comparison Tools
  11. Macbeath Regions and Hilbert Balls
  12. Fixed-Reference Comparison and Local Growth
  13. Polarity-Expansion Stability
  14. Minkowski Distortion Bound
  15. Hilbert Distortion Bound
  16. ...and 15 more sections

Key Result

Theorem 1

There exists an absolute constant $c \geq 1$ such that for any pair of convex bodies $G$ and $K$ in $\mathbb{R}^d$, with $O \in \mathop{\mathrm{int}}\nolimits(G)$ and $G \subset \mathop{\mathrm{int}}\nolimits(K)$, and any $\alpha \in (0,1]$,

Figures (5)

  • Figure 1: (a) The Funk distance and (b) the Hilbert distance.
  • Figure 2: The $\frac{1}{2}$-scaled Finsler balls in the (a) Funk and (b) Hilbert geometries (recentered on $x$).
  • Figure 3: Macbeath regions
  • Figure 4: Effect of translation on the polar.
  • Figure 5: (a) Busemann's pencil inequality, (b) complementary chords, and (c) Lemma \ref{['lem:comp-chord']}.

Theorems & Definitions (91)

  • Theorem 1: Duality for Hilbert-ball coverings
  • Theorem 2: Duality for boundary coverings by translates
  • Lemma 2.1: Projection-Section Duality
  • Lemma 2.2: Minkowski covering bound
  • Lemma 2.3: Hilbert covering bound
  • Lemma 2.4
  • proof
  • Lemma 2.5: Duality of Minkowski Holmes--Thompson measures
  • Lemma 2.6: Duality of Funk Holmes--Thompson measures
  • Lemma 2.7: Duality of Hilbert Holmes--Thompson measures
  • ...and 81 more