{"id":"91182c86-5064-46aa-902c-1441656f5a08","arxiv_id":"2607.29458","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every isotropic convex body in R^n, the mean width M*(K) is O(sqrt(n log n)), and the product M(K)M*(K) is O(log n), both optimal up to constants.","lead":"This paper proves that a high-dimensional convex body in a centered isotropic position has mean width at most a universal constant times sqrt(n log n), which is the optimal order. The proof uses a new heat-flow inequality connecting the support function averaged under Gaussian and uniform measures, plus recent dimension-free bounds on a third-moment parameter.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main theorem rests on the external dimension-free κ_n bound [31]; Section 2 is explicitly expository, so the proof of Theorem 1.1 is not self-contained and is conditional on an unverified external preprint.","rationale":"The reader's weakest assumption is exactly the right concern: Theorem 1.1 hinges on the right-hand side of Theorem 1.2, whose dimension-free form depends on the external κ_n bound from [31]. I checked the internal arguments: Theorem 1.3's heat-flow proof is correct, the Gaussian/spherical normalization (48) is standard, and the optimality examples are consistent. The Section 2 sketch of the κ_n bound is plausible and the algebra in Corollaries 2.6–2.7 checks out, but the paper explicitly disclaims originality there and leaves the approximation step to [11]. Therefore the central claim is conditionally supported rather than fully self-contained. This does not change the reader's CONDITIONAL verdict.","tokens_in":12237,"tokens_out":20171,"duration_ms":213445,"concrete_test":"Independently re-derive Theorem 2.5 from Proposition 2.4 with the full regularization argument: approximate a general centered log-concave μ by smooth log-concave measures with covariance Σ and verify that both N2=∫Tr(H^2)dν and Tr(Σ^2) pass to the limit, so Var(|X|^2)≤8Tr(Σ^2) holds without extra terms. If the approximation cannot be completed, the κ_n≤16 input and hence the proof of Theorem 1.1 collapse.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 is deduced from Theorem 1.3 by applying the right-hand side of Theorem 1.2, E h_K(X_K) ≤ C√log n E h_K(G). The paper does not prove the κ_n^2≤16 bound needed to remove the κ_n factor from the Eldan–Lehec/Bizeul–Klartag comparisons; it is imported from [31] and only sketched in Section 2, which the paper itself labels 'purely expository.' The sketch reduces κ_n≤4 to the anisotropic thin-shell estimate Var(|X|^2)≤8 Tr(Σ^2) (Theorem 2.5), but the proof of Theorem 2.5 is carried out under smoothness/regularity assumptions and the approximation argument is delegated to [11]. Thus the central claim is not supported by the paper alone: if [31]'s dimension-free bound is false, or if the anisotropic adaptation in §2.2 has a hidden assumption, Theorem 1.1 has no proof as written. This is a genuine conditionality, not an identified internal error: the new heat-flow inequality (Theorem 1.3) is internally sound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":12540,"tokens_out":16126,"duration_ms":150234,"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":[{"comment":"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.","section":"Section 2 / Theorem 1.1"}],"minor_comments":[{"comment":"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.","section":"Section 2, Corollary 2.7"},{"comment":"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.","section":"Section 2, Theorem 2.5"},{"comment":"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.","section":"Section 3.1, Theorem 1.3"},{"comment":"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.","section":"Section 3.3"},{"comment":"Typo: 'discusseion' should be 'discussion' in the paragraph before Notation.","section":"Introduction"},{"comment":"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.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper's main new theorem (Theorem 1.3) is sound, and the applications and optimality examples are convincing. The main risk is the dependence on the external preprint [31]. If the author can add a full proof of the κ_n bound (or if [31] becomes accepted and available to the referee), the paper would be acceptable. Given that Section 2 is explicitly expository and the central claim is conditional, I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The one thing you should know: the genuinely new step is Theorem 1.3, a short heat-flow inequality relating Gaussian and uniform means of the support function. The main theorem M*(K) ≤ C√(n log n) is derived cleanly from it, but the derivation relies on a dimension-free κ_n bound imported from an external preprint [31], and the paper's Section 2, which sketches that bound, is explicitly expository. So the result is probably right, but the paper as written is not self-contained on its load-bearing input.\n\nWhat's good: Theorem 1.3 is a real, checkable contribution. The proof uses the divergence theorem to get F'(s) ≤ n/2, chooses s* = 2W_K/n, and Jensen gives the inequality. I checked the algebra and it works. The deduction of Theorem 1.1 from Theorems 1.2 and 1.3 is also clean. The paper is honest about what is expository and what is new. The optimality examples (cube, cross-polytope, product) are standard but correctly assembled, and Corollary 1.5 extending Pisier's MM* bound to non-symmetric bodies in isotropic position is a nice payoff. Banach-Mazur and flatness consequences are straightforward but useful.\n\nSoft spots: The load-bearing κ_n^2 ≤ 16 is not proved in this paper. Section 2 is a sketch of Chen-Klartag's argument with an anisotropic twist, and the approximation argument removing regularity assumptions is delegated to [11]. If [31]'s bound turns out false, or if the anisotropic adaptation in §2.2 hides an assumption, Theorem 1.1 has no proof as written. The reader's conditional verdict is the right one. That said, this is a genuine conditionality, not an internal contradiction. The paper flags it clearly, so no one is being misled.\n\nWho it's for: anyone working in asymptotic convex geometry, especially on mean-width, MM* estimates, and isotropic positions. It deserves a serious referee: the new step is worth checking carefully, and the dependence on [31] should be verified. If [31] holds, this settles the optimal order of the mean-width. I'd send it to someone who can assess both the heat-flow argument and the thin-shell input.\n\nRecommendation: engage with it. Accept for review; the authors should be asked to either include a full proof of the κ_n bound or state the theorem as conditional on [31] and make the dependency explicit. It's a solid paper, but not fully self-contained yet.","headline":"Clean new heat-flow inequality; main theorem is conditional on an external κ_n bound the paper doesn't fully prove — still worth refereeing.","tokens_in":13042,"tokens_out":2566,"would_cite":true,"duration_ms":24769,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A40","60E15","46B06"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["mean-width","isotropic convex body","MM* estimate","heat-flow inequality","support function","log-concave vectors","thin-shell estimate","Pisier's estimate"],"falsifier":"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.","tokens_in":12116,"feed_emoji":"📐","tokens_out":3979,"duration_ms":41543,"temperature":0.7,"pith_summary":"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.","feed_headline":"Sharp mean-width bound M*(K) ≤ C√(n log n) proved","feed_subtitle":"Heat-flow inequality yields the optimal M* bound and extends Pisier's MM* estimate to nonsymmetric bodies.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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}."],"forward_implications":["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."],"fun_headline_variants":["Heat-flow inequality proves optimal M* bound","Optimal mean-width bound: M*(K) ≤ C√(n log n)","Pisier's MM* estimate now holds for nonsymmetric bodies","Sharp M* via new Gaussian-uniform support inequality","Dimension-free κ_n yields optimal M* and MM* bounds"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Heat-flow inequality proves optimal M* bound","Optimal mean-width bound: M*(K) ≤ C√(n log n)","Pisier's MM* estimate now holds for nonsymmetric bodies","Sharp M* via new Gaussian-uniform support inequality","Dimension-free κ_n yields optimal M* and MM* bounds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000194,"raw_usage":{"total_tokens":1179,"prompt_tokens":720,"completion_tokens":459,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":464,"completion_tokens_details":{"reasoning_tokens":374}},"tokens_in":464,"tokens_out":459,"duration_ms":4465,"temperature":1.0,"reasoning_tokens":374,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T06:43:50.466802+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}