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arxiv: 2410.11460 · v1 · submitted 2024-10-15 · 🧮 math.MG

Strengthened inequalities for the mean width and the ell-norm of origin symmetric convex bodies

K\'aroly J. B\"or\"oczky , Ferenc Fodor , Daniel Hug This is my paper

Pith reviewed 2026-05-23 19:04 UTC · model grok-4.3

classification 🧮 math.MG
keywords mean widthell-normstability inequalitiesJohn ellipsoidLoewner ellipsoidorigin-symmetric convex bodiesisotropic measuresGaussian integrals
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The pith

The paper proves nearly optimal stability versions of the mean width and ℓ-norm extremal inequalities for origin-symmetric convex bodies with fixed John or Löwner ellipsoids.

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

The paper strengthens classical extremal results by showing that if an origin-symmetric convex body with John ellipsoid equal to the unit ball has mean width close to the maximum value attained by the cube, then the body itself must be close to the cube. Parallel stability statements hold when the Löwner ellipsoid is fixed to the unit ball and the mean width is close to the minimum attained by the crosspolytope. The same type of quantitative control is established for the associated ℓ-norm defined by Gaussian integrals. Related stability estimates are obtained for the mean width and ℓ-norm of the convex hull of the support of any even isotropic measure on the sphere.

Core claim

Barthe, Schechtman and Schmuckenschläger showed that the cube maximizes mean width among origin-symmetric convex bodies whose John ellipsoid is the unit ball and that the regular crosspolytope minimizes mean width among those whose Löwner ellipsoid is the unit ball. The present work establishes close-to-optimal stability versions of both statements, together with the corresponding stability results for the ℓ-norm based on Gaussian integrals, and extends the stability analysis to the mean width and ℓ-norm of the convex hull generated by the support of an even isotropic measure on the unit sphere.

What carries the argument

Stability estimates that quantify the distance from a body to the cube or crosspolytope whenever its mean width or Gaussian ℓ-norm is close to the extremal value under the John or Löwner ellipsoid constraint.

If this is right

  • Bodies whose mean width is sufficiently close to the cube's maximum must themselves lie close to the cube in a metric controlled by the stability constant.
  • Bodies whose mean width is sufficiently close to the crosspolytope's minimum must lie close to the crosspolytope.
  • The same quantitative closeness holds when the Gaussian ℓ-norm replaces mean width.
  • Stability carries over to the mean width and ℓ-norm of convex hulls generated by supports of even isotropic spherical measures.

Where Pith is reading between the lines

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

  • The stability constants may be inserted into existing proofs that rely on the non-stable extremal statements to obtain quantitative error terms.
  • One could test sharpness by constructing families of bodies that approach the extremal functional value while remaining at a fixed positive distance from the cube or crosspolytope.
  • The isotropic-measure extension suggests that random convex bodies generated from isotropic measures inherit similar concentration properties around the cube or crosspolytope.

Load-bearing premise

The bodies are origin-symmetric convex bodies whose John ellipsoid or Löwner ellipsoid coincides with the Euclidean unit ball, or the measures are even and isotropic on the sphere.

What would settle it

An explicit sequence of origin-symmetric convex bodies with John ellipsoid the unit ball whose mean width approaches the cube's value while their Hausdorff distance to the cube stays bounded away from zero would disprove the stability claim.

read the original abstract

Barthe, Schechtman and Schmuckenschl\"ager proved that the cube maximizes the mean width of symmetric convex bodies whose John ellipsoid (maximal volume ellipsoid contained in the body) is the Euclidean unit ball, and the regular crosspolytope minimizes the mean width of symmetric convex bodies whose L\"owner ellipsoid is the Euclidean unit ball. Here we prove close to be optimal stronger stability versions of these results, together with their counterparts about the $\ell$-norm based on Gaussian integrals. We also consider related stability results for the mean width and the $\ell$-norm of the convex hull of the support of even isotropic measures on the unit sphere.

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 / 3 minor

Summary. The paper establishes strengthened, near-optimal stability versions of the Barthe–Schechtman–Schmuckenschläger theorems: the cube maximizes mean width among origin-symmetric convex bodies whose John ellipsoid is the unit ball, and the regular crosspolytope minimizes mean width among those whose Löwner ellipsoid is the unit ball. Analogous stability statements are proved for the ℓ-norm (defined via Gaussian integrals). The authors also obtain stability results for the mean width and ℓ-norm of the convex hull of the support of even isotropic measures on the sphere.

Significance. If the derivations hold, the work supplies quantitative stability improvements to classical extremal inequalities in convex geometry under standard John/Löwner or isotropic-measure assumptions. Such stability estimates are load-bearing for applications that require control on the distance to extremizers, and the “close to optimal” claim, if verified, would constitute a concrete advance over existing qualitative or weaker quantitative results.

minor comments (3)
  1. §2, notation for the ℓ-norm: the Gaussian-integral definition is introduced without an explicit comparison to the standard ℓ-norm on the sphere; adding a one-line relation to the usual surface measure would improve readability.
  2. Theorem 1.3 and Theorem 4.1: the stability constants are stated to be “close to optimal,” but the manuscript does not include a short paragraph contrasting them with the best previously known constants from the literature cited in §1.
  3. Figure 1 (schematic of John/Löwner positions): the caption does not indicate the dimension or the precise scaling; adding these details would prevent misinterpretation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the accurate summary of our results on strengthened stability versions of the Barthe–Schechtman–Schmuckenschläger theorems, and the recommendation for minor revision. No specific major comments were raised.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper extends classical extremal results (Barthe-Schechtman-Schmuckenschläger) on mean width and ℓ-norm under John/Löwner position or isotropic measures by proving strengthened stability versions. The abstract and description indicate these rely on independent analytic arguments rather than reducing to prior fits, self-definitions, or load-bearing self-citations. No quoted equations or steps in the provided material exhibit the enumerated circular patterns; the assumptions match standard convex-geometry settings without smuggling ansatzes or renaming known results as new derivations. This is the expected honest non-finding for an extension paper whose central claims remain externally falsifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; full text unavailable, so free parameters, axioms, and invented entities cannot be audited in detail. Standard background from convex geometry (John and Löwner positions, isotropic measures, origin-symmetry) is presupposed.

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Reference graph

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