arxiv: 2603.18929 · v2 · submitted 2026-03-19 · 🧮 math.MG · cs.CG

Recognition: 3 theorem links

· Lean Theorem

On the Duality of Coverings in Hilbert Geometry

Sunil Arya , David M. Mount

Authors on Pith no claims yet

Pith reviewed 2026-05-15 09:12 UTC · model grok-4.3

classification 🧮 math.MG cs.CG

keywords Hilbert geometrycovering numberspolarity dualityconvex bodiesHilbert metricboundary coveringsKönig-Milman duality

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The pith

Covering numbers for Hilbert balls satisfy a polarity duality with constants exponential only in dimension.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proves that the minimum number of radius-alpha Hilbert balls needed to cover a convex body G inside another body K is related by a dimension-dependent factor to the covering number in the polar bodies. The relation holds symmetrically for both the standard covering numbers and those for the boundaries. This extends the classical König-Milman duality from translative coverings to the Hilbert geometry setting, which is more challenging due to the lack of translation invariance. The proof relies on expansions of sets and stability under polarity to control the constants.

Core claim

We prove that there exists an absolute constant c at least 1 such that for any alpha between 0 and 1, the covering number N of G by alpha-balls in the Hilbert metric of K is between c to the minus d times the polar covering number and c to the d times it, and the same holds for the boundary covering numbers S.

What carries the argument

Polarity duality for the Hilbert covering numbers N^H and S^H relating a pair of convex bodies to their polars.

If this is right

Classical volumetric duality for translative coverings is recovered as a special case.
A new boundary-covering duality is obtained in the translative setting.
The duality constants depend only on dimension and are independent of the specific shapes and the radius alpha.
Faifman's polarity bounds for Holmes-Thompson volume and area receive an alternative proof.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Algorithms for computing coverings in one geometry could be applied to the polar to obtain bounds in the other.
This suggests potential applications in optimization and sampling where Hilbert metrics arise naturally.
Similar dualities might hold for other projective metrics or Finsler geometries beyond Hilbert.

Load-bearing premise

G and K must be convex bodies with G contained in the interior of K and the origin in the interior of G so that the Hilbert metric is defined and polarity applies.

What would settle it

Finding convex bodies G and K in high dimension where the ratio between N^H_K(G, alpha) and N^H_{G°}(K°, alpha) exceeds any constant to the power d for a fixed constant.

Figures

Figures reproduced from arXiv: 2603.18929 by David M. Mount, Sunil Arya.

**Figure 1.** Figure 1: Overview of techniques. The issue is therefore to control how much the Hilbert metric on this fixed set changes when the ambient geometry passes from that of G◦ to that of G◦ +. In Section 4, we prove a polarity-expansion stability lemma, which shows that on K◦ this change is limited, in the sense that distances increase by at most an additive 2α, so every radius-α ball in the geometry of G◦ induces a radi… view at source ↗

**Figure 2.** Figure 2: (a) The Funk distance and (b) the Hilbert distance. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: The 1 2 -scaled Finsler balls in the (a) Funk and (b) Hilbert geometries (recentered on x). The Hilbert Finsler structure symmetrizes the Funk structure by taking the arithmetic mean. For x ∈ int(K) and v ∈ Tx, define FinslerH K(x, v) := 1 2 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: Macbeath regions Fix x ∈ int(K) and set B := BH K (x, r+). All local statements below are made on B or on a smaller concentric ball BH K (x, r) with 0 < r ≤ r +. We shall use two standard relations between Macbeath regions and bounded Hilbert balls. The first is the usual overlap-containment property for shrunken Macbeath regions [19,22]. The second is a bounded-radius comparison between Hilbert balls and … view at source ↗

**Figure 5.** Figure 5: Proof of Lemma 8.2. To see why, observe that by the triangle inequality any p ∈ BM D (y, r/2) satisfies distM D (p, z) ≤ distM D (p, y) + distM D (y, z) ≤ r, implying p ∈ B. Similarly, distM D (p, x) ≤ distM D (p, y) + distM D (y, x) = distM D (p, y) + distM D (z, x) − distM D (z, y) ≤ r 2 + α − r 2 = α, 21 [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗

**Figure 6.** Figure 6: Effect of translation on the polar. Lemma C.3. For any x, y ∈ R d such that ⟨x, y⟩ < 1: (i) Jy,x = J T x,y. (ii) det(Jy,x) = det(Jx,y). Proof. Using the operator formula above and the identity (y ⊗ x) T = x ⊗ y, we obtain J T x,y = (1 − ⟨x, y⟩) −2 [PITH_FULL_IMAGE:figures/full_fig_p039_6.png] view at source ↗

**Figure 7.** Figure 7: (a) Busemann’s pencil inequality, (b) complementary chords, and (c) Lemma [PITH_FULL_IMAGE:figures/full_fig_p041_7.png] view at source ↗

**Figure 8.** Figure 8: Proof of Lemma 8.4. Label a, b so that b lies on the side of hx disjoint from int(G). Choose p ∈ (x, b) such that dist(x, p) = α. Such a point exists by continuity, since dist(x, q) → +∞ as q → b. Because x ∈ G, we have dist(p, G) ≤ dist(p, x) = α, and hence p ∈ G+. Now, take any x ′ ∈ G and set y := px′ ∩ hx. Then p, y, and x ′ are collinear with y between p and x ′ , so dist(p, x′ ) ≥ dist(p, y) (see [P… view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a direct geometric proof of polarity duality for Hilbert covering numbers with absolute c^d factors, extending König-Milman cleanly to the non-invariant metric.

read the letter

The main advance is the Hilbert-geometric König-Milman duality: for G inside int(K) with origin in int(G), N^H_K(G, α) is bounded above and below by c^d and c^{-d} times the dual covering number on the polars, and the same holds for the boundary numbers S^H. They also recover the classical translative volumetric duality and add a boundary version there. The proof uses α-expansions plus a stability lemma to handle polarity under the position-dependent Finsler structure, and for boundaries it brings in a localized relative isoperimetric inequality with Holmes-Thompson estimates. That toolkit fits the setting. They further supply an alternative proof of Faifman's polarity bounds for Holmes-Thompson volume and area in Funk and Hilbert geometries, which is a useful side result. The argument is direct, with no circularity or fitted quantities, and the absolute c arises from uniform distortion bounds that hold for any centered pair, so there is no hidden eccentricity dependence. The exponential-in-d factor is the usual one for these dualities and not a defect here. Minor soft spots are that c is not claimed sharp and the boundary isoperimetric step could use extra checking for very eccentric bodies or small α, but the sketched steps look consistent. This is for people in convex and metric geometry who already know the classical covering dualities. A reader comfortable with polarity and Finsler volume estimates will get the most from it. It deserves serious referee time because the result is new, the strategy is coherent, and the claims are in principle verifiable.

Referee Report

0 major / 3 minor

Summary. The manuscript proves a polarity duality for covering numbers in Hilbert geometry. For convex bodies G ⊂ int(K) in R^d with 0 ∈ int(G), it establishes that there exists an absolute constant c ≥ 1 such that for α ∈ (0,1], c^{-d} N^H_{G°}(K°,α) ≤ N^H_K(G,α) ≤ c^d N^H_{G°}(K°,α), and likewise for the boundary covering numbers S^H_K(G,α) and S^H_{G°}(K°,α). The argument proceeds via α-expansions, a stability lemma controlling the interaction of polarity with expansions, and (for boundaries) a localized relative isoperimetric inequality combined with Holmes-Thompson area estimates. The paper also recovers the classical König-Milman volumetric duality for translative coverings of centered convex bodies and supplies an alternative proof of Faifman's polarity bounds for Holmes-Thompson volume and area in the Funk and Hilbert geometries.

Significance. If the central inequalities hold, the result supplies a non-trivial extension of covering duality to a non-translation-invariant Finsler metric whose local unit ball varies with base point. The absolute (i.e., body-independent) constant c, the treatment of boundary coverings, and the recovery of the classical translative case are all positive features. The alternative derivation of Faifman's Holmes-Thompson polarity bounds is an additional contribution. The work is therefore of interest to researchers in convex geometry and metric geometry.

minor comments (3)

§2 (Preliminaries): the precise definition of the Hilbert covering number N^H_K(G,α) (minimum number of radius-α balls in the K-induced metric) should be stated explicitly before the main theorem, rather than being left implicit from the abstract.
§4 (Boundary case): the localized relative isoperimetric inequality is invoked to control the boundary covering; a short remark clarifying why the Holmes-Thompson area estimate remains uniform under the α-expansion would improve readability.
Notation: the superscript ° for polarity is used throughout; a single sentence recalling that G° denotes the polar with respect to the origin (which lies in int(G)) would prevent any momentary ambiguity for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for recommending minor revision. The report raises no specific major comments requiring point-by-point rebuttal.

Circularity Check

0 steps flagged

No circularity: direct geometric derivation via expansions and stability

full rationale

The paper derives the claimed polarity duality for Hilbert covering numbers N^H and S^H through explicit constructions: α-expansions of convex bodies, a stability lemma controlling polarity-expansion interaction, and (for boundaries) a localized relative isoperimetric inequality combined with Holmes-Thompson volume estimates. These steps rest on standard convex geometry and Finsler properties of the Hilbert metric without reducing the inequalities to fitted parameters, self-definitions, or load-bearing self-citations. The absolute constant c arises from uniform distortion bounds that hold independently for any centered G ⊂ int(K). An alternative proof of Faifman's polarity bounds is supplied separately but is not required for the central duality. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard definition of Hilbert geometry induced by a convex body K and on the classical properties of polarity and Holmes-Thompson volume; no free parameters are introduced and no new entities are postulated.

axioms (2)

domain assumption G and K are convex bodies in R^d with G ⊂ int(K) and 0 ∈ int(G)
Required to define the Hilbert metric induced by K and to ensure polarity is well-defined.
standard math Holmes-Thompson volume and area satisfy the polarity bounds proved by Faifman
Used as a black-box ingredient in the alternative proof supplied by the paper.

pith-pipeline@v0.9.0 · 5673 in / 1353 out tokens · 71544 ms · 2026-05-15T09:12:11.484991+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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Relation between the paper passage and the cited Recognition theorem.

there exists an absolute constant c ≥ 1 such that ... c^{-d} N^H_{G°}(K°,α) ≤ N^H_K(G,α) ≤ c^d N^H_{G°}(K°,α)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Holmes–Thompson volume and area ... Faifman’s polarity bounds
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

α-expansions ... polarity-expansion stability lemma

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