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arxiv: 2001.10706 · v1 · submitted 2020-01-29 · 🧮 math.MG

Strengthened inequalities for the mean width and the ell-norm

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

Pith reviewed 2026-05-24 15:05 UTC · model grok-4.3

classification 🧮 math.MG
keywords mean widthell-normstabilityregular simplexJohn ellipsoidLowner ellipsoidisotropic measuresconvex bodies
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The pith

The regular simplex uniquely extremizes mean width and the ell-norm under John and Lowner ellipsoid constraints, with quantitative stability.

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

The paper establishes stronger stability versions of known results that the regular simplex maximizes the mean width of convex bodies whose John ellipsoid is the unit ball, and equivalently maximizes the ell-norm when the Lowner ellipsoid is the unit ball. It likewise strengthens the minimization result for mean width under the Lowner ellipsoid condition. The stability quantifies that a body whose functional value is close to the extremal one must itself be close to the regular simplex in shape. Related stability statements are proved for the mean width and ell-norm of the convex hull of the support of any centered isotropic measure on the unit sphere.

Core claim

Stronger stability versions are proved for the extremal properties of the regular simplex in the mean width and ell-norm under the John and Lowner ellipsoid conditions, along with stability results for convex hulls of supports of centered isotropic measures.

What carries the argument

Quantitative stability estimates that bound the geometric distance of a convex body to the regular simplex by the deviation of its mean width or ell-norm from the extremal value.

If this is right

  • A convex body with John ellipsoid the unit ball and mean width close to the maximum must be close to the regular simplex.
  • The same closeness holds for the minimization case when the Lowner ellipsoid is fixed.
  • Analogous stability applies to the ell-norm under the corresponding ellipsoid normalization.
  • Quantitative stability holds for mean width and ell-norm when the body is the convex hull of the support of a centered isotropic measure.

Where Pith is reading between the lines

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

  • The stability estimates quantify the rate at which bodies approach the simplex as the functional value approaches its extremum.
  • Neighbouring extremal problems in convex geometry that use ellipsoid normalizations may admit comparable stability strengthenings.

Load-bearing premise

The convex bodies under consideration have their John ellipsoid equal to the Euclidean unit ball, or are formed as the convex hull of the support of a centered isotropic measure.

What would settle it

A convex body with John ellipsoid equal to the unit ball whose mean width is arbitrarily close to that of the regular simplex yet remains bounded away from the simplex in Hausdorff distance.

read the original abstract

Barthe proved that the regular simplex maximizes the mean width of convex bodies whose John ellipsoid (maximal volume ellipsoid contained in the body) is the Euclidean unit ball; or equivalently, the regular simplex maximizes the $\ell$-norm of convex bodies whose L\"owner ellipsoid (minimal volume ellipsoid containing the body) is the Euclidean unit ball. Schmuckenschl\"ager verified the reverse statement; namely, the regular simplex minimizes the mean width of convex bodies whose L\"owner ellipsoid is the Euclidean unit ball. Here we prove stronger stability versions of these results. We also consider related stability results for the mean width and the $\ell$-norm of the convex hull of the support of centered 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 proves quantitative stability strengthenings of Barthe's maximality theorem for the mean width of convex bodies whose John ellipsoid is the unit ball (equivalently, Schmuckenschläger's minimality result for the ℓ-norm under the Löwner ellipsoid normalization). It further establishes related stability inequalities for the mean width and ℓ-norm of the convex hull of the support of centered isotropic measures on the unit sphere, under the same normalizations.

Significance. If the stability estimates hold with explicit constants and sharp equality cases, the results provide useful quantitative versions of known extremal inequalities in convex geometry. Stability strengthenings of this type often enable applications to approximation problems and asymptotic estimates; the extension to isotropic spherical measures adds scope without introducing new normalizations.

minor comments (3)
  1. §2, after Definition 2.3: the transition from the John-position assumption to the stability estimate for mean width is stated without an explicit reference to the constant appearing in the preceding inequality; adding the dependence would clarify the quantitative nature of the result.
  2. Theorem 3.2 and the isotropic-measure case in §4: the notation for the ℓ-norm is reused from the convex-body setting without a separate definition; a brief reminder of the precise definition in the measure context would avoid ambiguity.
  3. Figure 1 (if present) or the equality-case discussion in §1: the caption or surrounding text does not explicitly state whether the depicted simplex achieves equality in all stated inequalities simultaneously.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. The referee's summary accurately captures the main contributions regarding stability strengthenings of Barthe's theorem and related results for isotropic measures. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation builds on external prior results

full rationale

The paper derives stability strengthenings of Barthe's maximality result for mean width (under John ellipsoid normalization) and Schmuckenschläger's minimality result (under Löwner ellipsoid normalization), plus analogous statements for convex hulls of centered isotropic measures. These rest on standard external normalizations and cited extremal theorems whose authors do not overlap with the present paper. No equations reduce the new stability estimates to fitted parameters or self-definitions by construction, no load-bearing self-citations appear, and no ansatz or uniqueness claim is imported from the authors' own prior work. The central claims remain independent of the inputs and are self-contained against the referenced external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review yields no visible free parameters, invented entities, or ad-hoc axioms beyond standard convexity and ellipsoid theory.

axioms (2)
  • standard math Existence and uniqueness of John and Löwner ellipsoids for convex bodies in Euclidean space
    Invoked implicitly by the statement that the ellipsoids are fixed to the unit ball.
  • standard math Definition and properties of centered isotropic measures on the sphere
    Used for the second family of stability results.

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