{"id":"603d1478-4a42-495d-8a43-9ed86c73df0a","arxiv_id":"2608.07216","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For origin-symmetric convex bodies in isotropic position, deterministic geometric arguments yield M(K) ≤ C log(n)/√n and M*(K) ≤ C√n log(n), hence MM* ≤ C log² n.","lead":"This paper provides new deterministic geometric proofs of near-optimal average-distance bounds for high-dimensional symmetric convex bodies. It matters because it replaces probabilistic machinery with explicit spectral geometry, relying only on the slicing theorem as a black box.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The paper's stated goal is to give deterministic geometric proofs of near-optimal mean-gauge and mean-width bounds, assuming only the slicing theorem through its small-ball consequence (1). The reader's verdict identified the small-ball estimate as the load-bearing premise; my stress-test agrees that this is the only high-dimensional input, but found no flaw in how it is used. I verified the two codimension estimates that constitute the core of the proofs: in Theorem 2.1, the weighted-covariance argument produces a subspace of codimension O(p) on which the p-th summand has curvature ≥ cp, by applying (1) to the isotropic log-concave projection P_F X; in Proposition 3.1, the same estimate is applied to the whitened tilted marginal, yielding a spectral bound #{λ_i(Cov(μ_ξ)) > D} ≤ 2Λ(ξ). The algebraic derivations in (9), (20), and the Legendre duality steps are consistent. The eigenvalue-counting and inverse-trace arguments are valid, and the comparison K ⊆ C S in Theorem 4.1 follows from the cited Klartag–Milman centroid–Laplace equivalence and standard inclusions. The proofs explicitly disclaim any independent proof of slicing, and the conditional bounds in Remark 4.3 match the expected L_n dependence. The only residual issue is the usual omission of a regularization argument for non-smooth auxiliary bodies, which is a standard fix and not a substantive gap. I therefore see no reason to change the ACCEPT verdict.","tokens_in":10452,"tokens_out":44677,"duration_ms":419391,"concrete_test":"Re-derive the two spectral lemmas—the codimension bound in the weighted-covariance argument of Theorem 2.1 and the codimension bound in Proposition 3.1—starting from the weaker small-ball estimate (2) instead of (1), and check that the resulting intermediate bounds match the L_n-dependent equations (9.1) and (9.3) in Remark 4.3. If they do, then (1) is exactly the input used and no hidden high-dimensional estimate enters; if an extra L_n factor or a change in the codimension exponent appears, the proof would have silently used a stronger input.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central estimates rest on the dimension-free small-ball estimate (1), a consequence of the slicing theorem. I checked the two places where the estimate is used: the codimension-O(p) weighted-covariance bound preceding (10), and the tilted-covariance bound in Proposition 3.1. In both, the measure to which (1) is applied is either an isotropic log-concave marginal or a whitened exponential tilt, and the resulting codimension bounds are r ≤ Cp and r ≤ 2p respectively. The dyadic eigenvalue-counting, inverse-trace, and spherical-Laplacian steps are internally consistent, and the Legendre-duality steps match the stated inequalities. I found no circularity or silent use of a stronger estimate, and Remark 4.3 correctly tracks the L_n factors if (1) is replaced by (2). The only caveat is the implicit C^2 regularity of the auxiliary bodies R and L_p(K); this is a standard approximation issue in convex geometry and does not undermine the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves deterministic geometric bounds for the mean gauge and the mean width of origin-symmetric convex bodies in probabilistic isotropic position. The two main results, Theorem 2.1 and Theorem 4.1, give M(K) ≤ C log(n)/√n and M*(K) ≤ C√n log(n), hence M(K)M*(K) ≤ C log² n. The first proof uses a quadratic aggregate of dyadic centroid bodies; the second uses a weighted aggregate of Laplace bodies. The only high-dimensional input is the dimension-free small-ball estimate (1), a consequence of the slicing theorem. Remark 4.3 records explicitly what the arguments give without slicing, replacing absolute constants by the hereditary quantities L_M(K) and L_{M*}(K).","tokens_in":10531,"tokens_out":14577,"duration_ms":137728,"significance":"The estimates themselves are not optimal: the mean-gauge bound was already known with the same order and the mean-width bound was recently improved to optimal up to constants. The value of the paper lies in the method. The proofs avoid stochastic localization, heat flow, and the Milman–Pisier theorem, and they isolate precisely where the slicing theorem enters. The curvature formulas (9) and (20), the spectral small-ball arguments, and the inverse-trace estimates are derived rather than asserted. Remark 4.3 is particularly valuable because it makes the dependence on lower-dimensional isotropic constants fully explicit. The extension to non-symmetric bodies in Remark 4.4 is a useful addition.","major_comments":[],"minor_comments":[{"comment":"The passage from the per-scale bound #{i : λ_i(H(u)) < 1 + cp} ≤ Cp to the uniform counting estimate N_u(t) ≤ Ct for all t ≥ 1 is compressed into a single sentence. Please spell out the dyadic matching for t near 1 and for t of order n, since this step is load-bearing for the inverse-trace estimate (8).","section":"§2, after Eq. (10)"},{"comment":"The Hessian computations assume C² smoothness of the auxiliary bodies R, L_p(K), and S, while the hypotheses only give convexity. Please add a standard approximation or regularization remark, or state explicitly that the Hessian identities are understood through approximation, so that the curvature arguments are fully justified at the level of the original body.","section":"Theorems 2.1 and 4.1"},{"comment":"The displayed estimates in Remark 4.3 are labeled (9.1)–(9.3), but the numbering in the main text stops at (28). Please renumber these displays to avoid confusion.","section":"Remark 4.3"},{"comment":"The inclusions K ⊆ C Z_n(K) ⊆ C(n/p) Z_p(K) are invoked as 'standard' without proof or reference. A citation to [7] or a one-line explanation would make the derivation of (14) easier to verify.","section":"§3, after Eq. (13)"},{"comment":"The phrase 'the trivial bound by n for t ≳ n' is terse. Please make explicit that for t of order n one uses N_u(t) ≤ n ≤ Ct, and that the remaining range of t is covered by choosing a dyadic scale p comparable to t.","section":"§2, proof of Theorem 2.1"}],"recommendation":"minor_revision","confidential_remarks":"This is a carefully written paper. I found no circularity: the slicing theorem and the Klartag–Milman centroid–Laplace equivalence are used as external inputs and are clearly identified as such. The main proofs are internally consistent, and Remark 4.3 gives a correct account of what happens without the slicing input. The remaining issues are local and can be addressed by adding explanatory details. The paper is within the scope of math.MG and makes a solid methodological contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, it does exactly what it says: it gives deterministic geometric proofs of M(K) ≤ C log(n)/√n and M*(K) ≤ C√n log(n) for isotropic origin-symmetric bodies, avoiding stochastic localization, martingale inequalities, and heat flow. Second, those bounds are strictly weaker than what Bizeul–Klartag and Bizeul already had; the point is the technique, not the numbers.\n\nWhat is actually new is the combination of dyadic centroid bodies with a quadratic aggregate and a weighted aggregate of Laplace bodies, plus a spectral eigenvalue-counting argument. That is real. The curvature formulas (9) and (20) are derived, the inverse-trace step (28) comes out of an honest layer-cake, and the small-ball input is used only through the slicing theorem in the dimension-free form (1). The paper is also scrupulous about tracking what happens without slicing: Remark 4.3 spells out the L_n factors and the hereditary quantities L_M(K) and L_M*(K), and says plainly that this is not an independent proof of slicing. That is good practice.\n\nThe soft spots are minor. Standard inclusions for centroid bodies are brushed over, the C^2 regularity of the auxiliary bodies R and L_p(K) is implicit, and the non-symmetric extension in Remark 4.4 is sketched rather than written out. None of these undermines the main proof. The paper is not machine-checked, so there is the usual residual risk of a hidden constant or index error, but I did not find one, and the stress-test note confirms the codimension counts.\n\nThis is a paper for people who work in asymptotic geometric analysis. It will not change the record books, but it shows a genuinely different route to near-optimal estimates and clarifies the role of the slicing theorem. I would send it to a serious referee. I might suggest the authors expand the regularity footnote and the non-symmetric remark, but I would not hold the paper up over them.","headline":"A careful deterministic proof of near-optimal M-M* bounds that trades a logarithmic factor for a clean geometric argument; the technique is the contribution, not the record.","tokens_in":11109,"tokens_out":1799,"would_cite":true,"duration_ms":21579,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A20","46B06","52A38"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves deterministic geometric bounds $M(K) \\leq C \\log(n)/\\sqrt{n}$ and $M^*(K) \\leq C\\sqrt{n}\\log(n)$ for origin-symmetric convex bodies in isotropic position, so $M(K)M^*(K) \\leq C\\log^2 n$.","keywords":["convex bodies","isotropic position","mean gauge","mean width","centroid bodies","Laplace bodies","small-ball estimates","slicing problem"],"falsifier":"Compute, for an explicit family of isotropic log-concave measures, the centered small-ball probability $\\nu(\\varepsilon\\sqrt{r}\\,B_2^r)$ for all $\\varepsilon>0$; if any measure exceeds $(C\\varepsilon)^r$ with the universal constant required in the proof, then the codimension control in the weighted-covariance lemma and in the tilted-covariance estimate breaks down, and the theorems' conclusions would not follow. A direct disproof would be a sequence of isotropic origin-symmetric bodies with $M(K)M^*(K)$ growing faster than $C\\log^2 n$.","tokens_in":10196,"feed_emoji":"📐","tokens_out":9033,"duration_ms":84955,"temperature":0.7,"pith_summary":"This paper proves that an origin-symmetric convex body in $\\mathbb{R}^n$ whose uniform probability measure is isotropic has mean gauge $M(K) \\leq C \\log(n)/\\sqrt{n}$ and mean width $M^*(K) \\leq C\\sqrt{n}\\log(n)$. Multiplying the two inequalities gives $M(K)M^*(K) \\leq C\\log^2 n$, missing the conjectured optimal order $O(\\log n)$ by only one logarithmic factor. The point is not just the numbers but the method: both estimates come from deterministic geometric arguments over dyadic scales, using centroid and Laplace bodies, spectral counting, Legendre duality, and the spherical Laplacian, with no stochastic localization or heat flow. The only high-dimensional input is the dimension-free small-ball estimate that follows from the slicing theorem, and without that input the same arguments yield bounds carrying an isotropic-constant factor.","feed_headline":"Isotropic bodies get M M* ≤ C log² n","feed_subtitle":"Deterministic dyadic curvature arguments prove both bounds without stochastic methods.","key_machinery":"Centroid bodies are defined by $h_{Z_p(K)}(u)=(E|\\langle X,u\\rangle|^p)^{1/p}$, and Laplace bodies by $L_p(K)=p\\{\\Lambda_K\\le p\\}^{\\circ}$, where $\\Lambda_K$ is the logarithmic Laplace transform of the uniform measure; the two families are comparable by a known theorem, which is what makes the Laplace-body route possible. The main mechanism is dyadic aggregation: at each scale $p$ a curvature estimate holds only outside a subspace of codimension $O(p)$, and by summing over dyadic scales with weights and applying the min-max principle and the layer-cake formula, the paper extracts global spectral information, namely an inverse-trace bound. Legendre duality turns upper or lower curvature bounds on one body into the complementary bounds on its polar, and the spherical Laplacian identity converts the inverse-trace bound into the desired spherical $L^2$ estimate.","core_discovery":"The central claim is that both the isotropic mean gauge and the isotropic mean width can be bounded by deterministic geometry at the orders $\\log n/\\sqrt{n}$ and $\\sqrt{n}\\log n$. For $M(K)$, the proof forms a body whose support function squares as $|u|^2$ plus the sum over dyadic $p$ of centroid-body moment terms; a weighted-covariance lemma shows that each dyadic summand has Hessian curvature of order $p$ outside a subspace of codimension $O(p)$, and a dyadic eigenvalue-counting argument converts this into a logarithmic inverse-trace bound. Legendre duality and the spherical Laplacian identity then transfer that bound to the spherical average of the gauge. For $M^*(K)$, the same machine is run on the Laplace bodies associated to the logarithmic Laplace transform; a spectral estimate for tilted covariance matrices supplies the curvature, the dyadic aggregation yields an inverse-trace bound of order $n^2\\log^2 n$ on the dual gauge, and the spherical integration gives the mean-width estimate. The arguments are stated for origin-symmetric bodies and extended to centered non-symmetric bodies.","pith_inferences":["The dyadic-aggregation template should apply to other rotationally averaged functionals of log-concave measures whenever one can prove a codimension-$O(p)$ curvature estimate at each scale; $M$ and $M^*$ are the two natural test cases.","Because the small-ball estimate is the only high-dimensional input, any sharpening of the small-ball bound for centered balls would automatically improve constants in these theorems without changing the geometric argument.","The Laplace-body formulation suggests that mean-width control is essentially a spectral statement about tilted covariance matrices, so the same estimates may transfer to families of tilted measures and to variance bounds for log-concave measures.","A natural check is whether the dyadic layer-cake step is what loses the extra logarithm in the $MM^*$ product, by optimizing the aggregation weights or replacing the dyadic sum with a continuous one."],"forward_implications":["Every origin-symmetric isotropic body satisfies $M(K)M^*(K) \\leq C\\log^2 n$, within one logarithmic factor of the optimal order.","The mean-width bound $M^*(K) \\leq C\\sqrt{n}\\log(n)$ improves the previous mean-width estimate by one logarithmic factor, while avoiding the heavier functional-analytic ingredients of earlier approaches.","The mean-gauge argument reaches the optimal order $\\log n/\\sqrt{n}$ by a purely Euclidean route, without stochastic localization, martingale inequalities, or heat flow.","Both estimates extend to centered non-symmetric bodies with only minor modifications to the one-sided centroid bodies or the Laplace-body construction.","Without the slicing theorem, the arguments yield bounds with an extra factor equal to the supremum of the isotropic constants of the relevant marginals, so the method does not give an independent proof of slicing."],"supporting_citations":[{"why":"Resolves the slicing conjecture and supplies the small-ball estimate (1), the only high-dimensional input used in both proofs.","marker":"[12]"},{"why":"Proves the comparability of Laplace bodies $L_p(K)$ and centroid bodies $Z_p(K)$, the basis for the mean-width argument.","marker":"[13]"},{"why":"Provides the logarithmic Berwald inequality used to lower-bound weighted covariances in the mean-gauge weighted-covariance lemma.","marker":"[5]"},{"why":"Gives the earlier mean-width bound that Theorem 4.1 improves by one logarithmic factor.","marker":"[15]"},{"why":"Establishes the optimal mean-gauge bound that the paper's first estimate reproduces up to a logarithmic factor by a direct geometric route.","marker":"[4]"},{"why":"Gives an alternative proof of the slicing theorem through small-ball estimates and can replace (1) in the arguments.","marker":"[2]"}],"fun_headline_variants":["Dyadic curvature bounds mean gauge and width","M M* ≤ C log² n proved via dyadic bodies","Mean width and gauge: deterministic logarithmic bounds","Isotropic convex bodies get log² n product bound","Centroid and Laplace bodies tame isotropic means"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is the dimension-free small-ball estimate that every isotropic log-concave measure in $\\mathbb{R}^r$ gives mass at most $(C\\varepsilon)^r$ to any ball of radius $\\varepsilon\\sqrt{r}$; if that estimate failed, the codimension-$O(p)$ curvature lemmas would fail and the bounds would degrade by a factor depending on isotropic constants.","fun_headline_variants_meta":{"raw":{"variants":["Dyadic curvature bounds mean gauge and width","M M* ≤ C log² n proved via dyadic bodies","Mean width and gauge: deterministic logarithmic bounds","Isotropic convex bodies get log² n product bound","Centroid and Laplace bodies tame isotropic means"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000274,"raw_usage":{"total_tokens":1699,"prompt_tokens":1066,"completion_tokens":633,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":682,"completion_tokens_details":{"reasoning_tokens":570}},"tokens_in":682,"tokens_out":633,"duration_ms":6439,"temperature":1.0,"reasoning_tokens":570,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T12:37:40.859621+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for an explicit family of isotropic log-concave measures, the centered small-ball probability $\\nu(\\varepsilon\\sqrt{r}\\,B_2^r)$ for all $\\varepsilon>0$; if any measure exceeds $(C\\varepsilon)^r$ with the universal constant required in the proof, then the codimension control in the weighted-covariance lemma and in the tilted-covariance estimate breaks down, and the theorems' conclusions would not follow. A direct disproof would be a sequence of isotropic origin-symmetric bodies with $M(K)M^*(K)$ growing faster than $C\\log^2 n$.","supporting_citations":[{"cited_title":"Klartag and J","cited_arxiv_id":null,"evidence_quote":"Resolves the slicing conjecture and supplies the small-ball estimate (1), the only high-dimensional input used in both proofs."},{"cited_title":"Klartag and E","cited_arxiv_id":null,"evidence_quote":"Proves the comparability of Laplace bodies $L_p(K)$ and centroid bodies $Z_p(K)$, the basis for the mean-width argument."},{"cited_title":"Borell,Convex measures on locally convex spaces, Ark","cited_arxiv_id":null,"evidence_quote":"Provides the logarithmic Berwald inequality used to lower-bound weighted covariances in the mean-gauge weighted-covariance lemma."},{"cited_title":"Milman,On the mean-width of isotropic convex bodies and their associated Lp- centroid bodies, Int","cited_arxiv_id":null,"evidence_quote":"Gives the earlier mean-width bound that Theorem 4.1 improves by one logarithmic factor."}],"review_version":1}