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arxiv: 2606.01082 · v1 · pith:GHRTXA5Inew · submitted 2026-05-31 · 🧮 math.MG

Dimension-free estimates for covering functionals of simplices and ell_p balls

Feifei Chen , Chan He , Senlin Wu This is my paper

Pith reviewed 2026-06-28 16:16 UTC · model grok-4.3

classification 🧮 math.MG
keywords covering functionalsconvex bodiessimplicescross-polytopesell_p ballsdimension-free estimatescovering numbers
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The pith

For n-simplices the covering factor Γ_{2^n} tends to 1/2 as dimension grows, while for all ℓ_p balls a uniform bound strictly below 1 holds in every dimension.

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

The paper defines and studies Γ_{2^n}(K), the smallest number γ such that the convex body K can be covered by 2^n translates of the scaled body γK. It proves that this quantity for the standard n-simplex approaches 1/2 in the high-dimensional limit. For the cross-polytope it establishes an explicit bound of 5/6 valid in every dimension together with a stricter limsup bound, and it shows that the same phenomenon of a dimension-independent upper bound less than 1 occurs uniformly across the entire family of ℓ_p balls. A reader would care because the results supply concrete dimension-free control on covering efficiency for these basic convex sets, which directly constrains how well they can be approximated or tiled by scaled copies.

Core claim

The paper establishes that Γ_{2^n}(Δ_n) tends to 1/2 as n tends to infinity. For the cross-polytope B_1^n it proves Γ_{2^n}(B_1^n) ≤ 5/6 for every n ≥ 2 and that the limsup as n → ∞ is at most 0.641…. It further proves there exists a constant κ_* < 1 such that Γ_{2^n}(B_p^n) ≤ κ_* holds simultaneously for all n ≥ 2 and all p in [1, ∞].

What carries the argument

Γ_{2^n}(K) is the infimum of all γ > 0 such that K is contained in a union of 2^n translates of γK; the proofs obtain upper bounds on this infimum via explicit geometric coverings and support-function or volume comparisons.

If this is right

  • The simplex covering problem admits coverings whose relative size approaches exactly one half in the limit.
  • Every cross-polytope admits a covering by 2^n scaled copies whose factor is at most 5/6, independent of dimension.
  • The limsup of the covering factor for cross-polytopes cannot exceed the stated numerical value 0.641….
  • A single number κ_* < 1 works as an upper bound for the covering factor of every ℓ_p ball in every dimension.

Where Pith is reading between the lines

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

  • The same style of covering argument might produce dimension-free bounds for other families of symmetric convex bodies beyond the ℓ_p balls.
  • If the limit 1/2 for simplices is sharp, then in high dimensions the optimal covering must use translates whose centers are distributed in a highly symmetric way relative to the simplex vertices.
  • The uniform bound κ_* suggests that the covering functional Γ_{2^n} stays bounded away from 1 for all bodies whose unit ball is an ℓ_p ball, which could be tested computationally in moderate dimensions.

Load-bearing premise

The explicit geometric constructions and the volume or support-function comparisons that produce the numerical bounds and the uniform κ_* remain valid without dimension-dependent deterioration as n increases.

What would settle it

An explicit covering construction or volume computation showing that Γ_{2^n}(Δ_n) stays above 0.6 for arbitrarily large n, or that Γ_{2^n}(B_p^n) exceeds 0.9 for some fixed p and a sequence of n tending to infinity.

Figures

Figures reproduced from arXiv: 2606.01082 by Chan He, Feifei Chen, Senlin Wu.

Figure 1
Figure 1. Figure 1: The event E(q) in Lemma 2.5. Lemma 2.5. Let E(q) be the event given by (cf [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The core mechanism behind Proposition 4.6. Proposition 4.6. There exists N∗ < ∞ such that Γ2n (B n p ) ≤ κ whenever n ≥ N∗ and p ∈ [π0, Qn(β)]. Proof. Set R = 3, S = S3(n, p), β0 = 9 5 , and r = 5 3 . Since 1 < r < β0 < β, we have δ := 1 − ln r ln β0 > 0. Claim 4.7. There exists N1 < ∞ such that, whenever n ≥ N1, p ∈ [π0, Qn(β)], and S ∈ S, we have, with d = n − #S, d ≥ 2n 3 and p ≤ ln d ln β0 . Proof of C… view at source ↗
read the original abstract

We study \(\Gamma_{2^n}(K)\), the least positive number \(\gamma>0\) such that an \(n\)-dimensional convex body \(K\) can be covered by \(2^n\) translates of \(\gamma K\). For \(n\)-simplices \(\Delta_n\), we prove that \(\Gamma_{2^n}(\Delta_n)\), as a sequence in \(n\), tends to \(1/2\). For the cross-polytope \(B_1^n\), we show that \(\Gamma_{2^n}(B_1^n)\leq5/6\) holds for all \(n\geq2\), and that \(\limsup_{n\to\infty}\Gamma_{2^n}(B_1^n)\leq0.641\cdots\). Finally, we prove the existence of a constant \(\kappa_*<1\) such that \(\Gamma_{2^n}(B_p^n)\leq\kappa_*\) for all \(n\geq2\) and all \(p\in[1,\infty]\).

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.

Referee Report

0 major / 2 minor

Summary. The manuscript studies the covering functional Γ_{2^n}(K), the smallest γ>0 such that an n-dimensional convex body K can be covered by 2^n translates of γK. It proves that Γ_{2^n}(Δ_n) tends to 1/2 as n→∞ for the standard n-simplex, that Γ_{2^n}(B_1^n)≤5/6 for all n≥2 with limsup_{n→∞}Γ_{2^n}(B_1^n)≤0.641… for the cross-polytope, and that there exists κ_*<1 such that Γ_{2^n}(B_p^n)≤κ_* for all n≥2 and all p∈[1,∞].

Significance. If the results hold, they supply explicit dimension-free upper bounds and a limit for covering functionals on simplices and ℓ_p balls. The explicit geometric constructions for the simplex and cross-polytope cases, together with the reduction of the general p-case to a compact family of extremal bodies whose covering constants remain bounded away from 1 uniformly in n and p, constitute a concrete contribution to asymptotic convex geometry.

minor comments (2)
  1. The numerical value 0.641… in the abstract and main statement for the limsup on B_1^n would benefit from an explicit reference to the section or computation (e.g., the linear-programming or recursive scheme) that produces it.
  2. Notation for the standard simplex Δ_n and the ℓ_p balls B_p^n should be recalled or referenced at the first use in the introduction for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's claims rest on explicit geometric constructions for simplices and cross-polytopes, together with support-function and volume-ratio arguments that are carried out with constants independent of dimension. No equations reduce by construction to fitted parameters or self-definitions, and no load-bearing self-citations or uniqueness theorems imported from the authors' prior work appear. The existence of κ_* follows from reduction to a compact family of extremal bodies with uniform positive margin away from 1. The derivation chain is self-contained against external geometric benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract; all claims are stated as proved statements about an existing functional.

pith-pipeline@v0.9.1-grok · 5705 in / 1175 out tokens · 31408 ms · 2026-06-28T16:16:31.923437+00:00 · methodology

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Reference graph

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