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A heat-flow inequality proves the optimal mean-width bound for isotropic convex bodies, M*(K) ≤ C√(n log n), and sharpens the MM* product estimate to C log n.

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2026-08-03 06:43 UTC pith:VTOST4CW

load-bearing objection Clean new heat-flow inequality; main theorem is conditional on an external κ_n bound the paper doesn't fully prove — still worth refereeing. the 1 major comments →

arxiv 2607.29458 v1 pith:VTOST4CW submitted 2026-07-31 math.MG math.FAmath.PR

Optimal MM^* bounds for convex bodies

Pierre Bizeul This is my paper
classification math.MG math.FAmath.PR MSC 52A4060E1546B06
keywords mean-widthisotropic convex bodyMM* estimateheat-flow inequalitysupport functionlog-concave vectorsthin-shell estimatePisier's estimate
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proves the optimal mean-width bound for convex bodies in isotropic position: M*(K) ≤ C√(n log n), matching the cross-polytope. The main new ingredient is a geometric inequality, obtained through a heat-flow argument, that ties the Gaussian average of the support function to its average over the body itself. Combined with an existing Gaussian-versus-log-concave comparison that now has a dimension-free constant, this yields the mean-width bound and, together with the known mean-norm estimate, gives the sharp MM* product bound M(K)M*(K) ≤ C log n. This extends Pisier's classical MM* estimate to non-symmetric convex bodies and to the isotropic position. A sympathetic reader sees the paper as completing a chain that makes the isotropic position simultaneously optimal for mean width, mean norm, and their product.

Core claim

For every centered convex body K in R^n, the paper proves (E h_K(G))^2 ≤ 2n E h_K(X_K), where G is a standard Gaussian vector, X_K is uniform on K, and h_K is the support function. This heat-flow inequality requires no isotropy and is the load-bearing new step. Applying it to an isotropic body and combining with the gauge-order comparison E h_K(X_K) ≤ C√(log n) E h_K(G) (which itself now holds with a universal constant thanks to the dimension-free bound on κ_n) yields M*(K) ≤ C√(n log n). The same combination with the mean-norm estimate M(K) ≤ C√(log n / n) gives the product bound ℓ(K) ≤ C log n, extending Pisier's MM* estimate to arbitrary convex bodies in isotropic position.

What carries the argument

The central identity is the heat-flow inequality of Theorem 1.3: (E h_K(G))^2 ≤ 2n E h_K(X_K), proved by running a Brownian motion from the uniform measure on K and using the fact that the heat semigroup of the support function has gradient in K, so the boundary term is controlled by the support function of the outward normal. The other ingredient is the gauge comparison theorem, which says for any gauge ∥·∥ and isotropic log-concave X, E∥X∥ is comparable to E∥G∥ up to √(log n) factors; its right-hand side now has a universal constant because the third-moment parameter κ_n satisfies κ_n ≤ 4, a dimension-free bound imported from an external preprint and only sketched here.

Load-bearing premise

The deduction of the mean-width theorem relies on the external dimension-free bound on the third-moment parameter κ_n (equivalently, the anisotropic thin-shell estimate Var(|X|^2) ≤ 8 Tr(Σ^2)), which the paper quotes from another preprint and only sketches; if that bound fails, the main theorem does not follow from this proof.

What would settle it

Construct a sequence of isotropic log-concave vectors X_n for which sup over gauges ∥·∥ of E∥X_n∥ / (√(log n) E∥G∥) tends to infinity, or directly exhibit a centered log-concave vector with covariance Σ such that Var(|X|^2) > 8 Tr(Σ^2) (or κ_n > 4). Either would falsify the imported dimension-free estimate that the proof needs.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The mean-width of any isotropic convex body is O(√(n log n)), and this is optimal since the isotropic cross-polytope attains the order.
  • The product M(K)M*(K) in isotropic position is O(log n), sharp even in the origin-symmetric class and even if one optimizes over affine positions.
  • Pisier's MM* estimate now holds for non-symmetric convex bodies, with the position chosen to be isotropic rather than a special linear image.
  • Banach–Mazur distances between convex bodies are bounded by C n log n, improving the previously known polylogarithmic bounds.
  • The flatness constant satisfies Flt(n) ≤ C n log n, improving the earlier O(n log^2 n) bound.

Where Pith is reading between the lines

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

  • The heat-flow inequality itself is position-free and self-contained; it may hold with better constants or extend to other homogeneous convex functions, offering a route to mean-width control without stochastic localization.
  • The proof of Theorem 1.1 is conditional on the imported dimension-free bound κ_n ≤ 16; if that bound fails, the mean-width theorem would still hold by the heat-flow inequality alone only up to an unknown n^ε factor, not the sharp √(log n).
  • The sharpness example Q_m × P_m suggests that to prove Conjecture 1.8 (√(log n) MM* for symmetric bodies) one must leave the isotropic position; the paper's own product example shows the isotropic position can force the larger log n factor.
  • The anisotropic thin-shell estimate Var(|X|^2) ≤ 8 Tr(Σ^2) may be of independent interest in high-dimensional probability, and the paper sketches how it also yields the KLS bound Ψ_n ≤ C (log n)^{1/4}.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 6 minor

Summary. The paper proves the optimal mean-width bound M*(K) ≤ C√(n log n) for isotropic convex bodies in R^n (Theorem 1.1). The main new ingredient is Theorem 1.3, a heat-flow inequality showing (E h_K(G))^2 ≤ 2n E h_K(X_K). Combining this with the gauge comparison of Eldan–Lehec and Bizeul–Klartag (Theorem 1.2), whose statements become dimension-free assuming a bound κ_n^2 ≤16, yields the mean-width estimate. The paper also derives the MM* product bound M(K)M*(K) ≤ C log n (Corollary 1.5), discusses sharpness and affine optimality, and records applications to Banach–Mazur distances and the flatness constant.

Significance. The mean-width estimate matches the lower bound from the cross-polytope and improves the previous O(√n log^2 n) bound of E. Milman. The heat-flow argument in Theorem 1.3 is elegant, self-contained, and of independent interest. If the external κ_n bound is valid, the paper resolves the mean-width problem in the isotropic position and gives the first MM* estimate for non-symmetric bodies in an explicit position, with optimal log n order. The optimality examples and applications are well chosen. The main weakness is that the proof of Theorem 1.1 is not self-contained: it relies on the dimension-free bound κ_n^2 ≤16 from the unpublished preprint [31], and Section 2 is explicitly expository.

major comments (1)
  1. [Section 2 / Theorem 1.1] The proof of Theorem 1.1 is conditional on the external bound κ_n^2 ≤16 (Corollary 2.7) from [31]. Section 2 is explicitly 'purely expository', and the anisotropic thin-shell estimate (Theorem 2.5) is proved only under smoothness assumptions, with the approximation step delegated to [11]. Since Theorem 1.2 and hence Theorem 1.1 require this dimension-free bound, the central claim is not supported by the present manuscript alone. Please either include a complete proof of the κ_n bound (including the approximation argument) in an appendix, or explicitly state Theorems 1.1, 1.2 and Corollary 1.5 as conditional on [31]. As written, a failure of [31] would invalidate the main theorem.
minor comments (6)
  1. [Section 2, Corollary 2.7] The inequality ||B||² = E(⟨X,θ⟩⟨BX,X⟩) ≤ Var(⟨BX,X⟩)^{1/2} is only true after centering: use E⟨X,θ⟩=0 to write the left side as E(⟨X,θ⟩(⟨BX,X⟩−Tr(B))). Please add this justification.
  2. [Section 2, Theorem 2.5] The passage from the regular to the general case of centered log-concave vectors is delegated to [11]; a brief explanation or a precise reference to the approximation argument would help the reader.
  3. [Section 3.1, Theorem 1.3] The commutation ∇P_s h_K = P_s(∇h_K) and the differentiation under the integral sign are used without comment; a sentence on the regularization argument would improve clarity.
  4. [Section 3.3] In the product example K=Q_m×P_m, the intermediate value E h_{P_m}(G_2) ≃ m√(log m) is later divided by E|G|; consider making the normalization explicit.
  5. [Introduction] Typo: 'discusseion' should be 'discussion' in the paragraph before Notation.
  6. [References] Reference [31] is an arXiv preprint; if the manuscript is intended for publication, the status of this reference should be updated or the dependence should be addressed as in Major Comment 1.

Circularity Check

0 steps flagged

No circularity: the central heat-flow inequality (Theorem 1.3) is proved from scratch, and the gauge comparison and κ_n bound are independent external inputs.

full rationale

The derivation chain is not circular. Theorem 1.3, the main new ingredient, is proved by a direct heat-flow argument (§3.1) and does not assume Theorem 1.1, Theorem 1.2, or any fitted value. The deduction of Theorem 1.1 in §3.2 combines Theorem 1.3 with the right-hand gauge comparison of Theorem 1.2, which is attributed to Eldan and Lehec [15]; the left-hand comparison from [7] is not used in that deduction. The dimension-free bound κ_n^2 ≤ 16 is imported from the external preprint [31] and only sketched in §2, which the paper itself labels 'purely expository and contains no original content.' This is a genuine conditionality—if [31] were false, Theorem 1.1 would lack support—but it is not circularity: [31] is not derived from the paper's own conclusions, and Theorem 1.3 is internally proved. The use of the author's own [7] for the mean-norm half of Corollary 1.5 and of [6] for the slicing step is also not circular: these are prior independent results with their own stated assumptions, not restatements of the present theorem. No parameter is fitted to the predicted quantities, and no uniqueness or ansatz is imported from a self-citation to force the main result. The paper is therefore best assessed as non-circular, with its reliance on external results noted as a limitation rather than a circular step.

Axiom & Free-Parameter Ledger

0 free parameters · 8 axioms · 0 invented entities

No free parameters are fitted and no new entities are introduced. The central theorem rests on a chain of external domain assumptions, the most load-bearing being the dimension-free kappa_n bound from [31] and the Eldan-Lehec gauge comparison. These are independent results, not derived in this paper; Section 2 is explicitly expository.

axioms (8)
  • domain assumption Right-hand gauge comparison for isotropic log-concave vectors: E||X|| <= C sqrt(log n) E||G|| for every gauge (Theorem 1.2, from Eldan-Lehec [15] with kappa_n)
    Used directly in the proof of Theorem 1.1; not proved in this paper.
  • domain assumption Dimension-free third-moment bound kappa_n^2 <= 16 (Eq. (14), from [31])
    Makes Theorem 1.2 dimension-free; Section 2 is expository and cites [31] and [11].
  • domain assumption Sharp thin-shell estimate Var(|X|^2) <= 8n for isotropic log-concave X (Chen-Klartag [11])
    Basis for the anisotropic adaptation in Section 2 and hence for the kappa_n bound.
  • domain assumption Slicing problem solution: for volume-normalized K, the isotropic scaling lambda_K is of order sqrt(n) ([27],[6])
    Used in Corollary 1.6 for affine optimality of M and M*.
  • domain assumption Giannopoulos-Milman characterization of minimal mean-width position (Eq. (51))
    Used to show the cube and cross-polytope are in minimal mean-width and mean-norm positions.
  • domain assumption Banaszczyk-Litvak-Pajor-Szarek lower bound ell(Delta) >= c log n for simplices
    Used for the lower bound in the affine-optimality statement of Corollary 1.7.
  • standard math Heat semigroup, divergence theorem, and support function properties used in the proof of Theorem 1.3
    Standard background invoked in Section 3.1 for the heat-flow inequality.
  • standard math Brascamp-Lieb inequality and moment-measure theorem in Section 2
    Standard tools in the expository sketch of the kappa_n bound.

pith-pipeline@v1.3.0-daily-deepseek · 25 in / 24773 out tokens · 370533 ms · 2026-08-03T06:43:50.466802+00:00 · methodology

0 comments
read the original abstract

Let $K\subset\R^n$ be a convex body in isotropic position. We prove the optimal mean-width estimate \[ M^*(K)\leq C\sqrt{n\log n}. \] The main new ingredient is a geometric inequality relating the Gaussian mean of the support function to its mean under the uniform measure on $K$, obtained through a heat-flow argument. Combined with the Gaussian-log-concave comparison of Eldan and Lehec and the newly available dimension-free bound on the third-moment parameter $\kappa_n$, this yields the result. The boundedness of $\kappa_n$ also makes the mean-norm estimate of Bizeul and Klartag sharp. Combining both estimates yields \[ M(K)M^*(K)\leq C\log n, \] extending Pisier's $MM^*$ estimate to non-symmetric convex bodies and to the isotropic position.

discussion (0)

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