{"id":"cdf066ef-2486-4b57-89cb-b2ee639ae92d","arxiv_id":"2607.29522","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every n-dimensional convex body can be linearly transformed so that its mean-width-to-volume ratio is at most a universal constant times sqrt(log n); the simplex and crosspolytope are extremal up to constants.","lead":"The paper proves that among all convex bodies in n-dimensional space, the simplex and crosspolytope are essentially the worst cases for mean width and covering entropy, up to universal constants. This settles a sharp bound that previously was known only conditionally on the KLS conjecture.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The universal constant in Theorem 3.1 rests on Bobkov's B-position lower bound α²_{K°} ≥ c (Prop 2.1(ii)); if this imported property fails, the comparison constant depends on K and the √log(en) order of Theorem 1.2 does not follow.","rationale":"The reader's weakest_assumption identified the same point: Bobkov's B-position properties, specifically the lower bound on α², are the crucial external input. My stress-test confirms this is the most load-bearing step because it is used exactly once, in Eq. (21), to make the comparison constant universal; without it, all three headline theorems degrade to dimension-dependent bounds. I also checked the surrounding argument: the stochastic-localization comparison (Propositions 3.2–3.3, Lemmas 3.4–3.8) appears mathematically sound up to minor typographical issues (Prop 3.2 coefficient, Prop 3.3 T display), and the difference-body reduction for non-symmetric K, although deferred, is routine via Rogers–Shephard and reverse Santaló. The computational test on ℓ_p balls is a reasonable sanity check, but the decisive check is verifying the cited proposition. Since the concern is a missing proof of an external result rather than a demonstrated contradiction, it does not change the reader's CONDITIONAL verdict; it strengthens the case that the condition (verification of Prop 2.1) should be supplied.","tokens_in":19368,"tokens_out":44533,"duration_ms":380590,"concrete_test":"Verify Proposition 2.1(ii) directly against Bobkov [10, Prop. 3.11] and re-derive it in the manuscript's notation: from the B-position first-order condition Cov(γ_K)=α²_K I and vol_n(K)=1, prove α²_K ≥ c with c independent of n. If the best available lower bound is dimension-dependent (e.g., c n^{-β}), recompute Eq. (21) with that bound; any n-dependence in C′ would destroy the universal √log(en) order. As a computational spot-check, for n=2,...,8 and K = scaled ℓ_p^n balls (normalized to volume 1), numerically optimize T↦γ_n(TK) over SL(n) and compute α²(TK); if α² dips below a fixed positive constant as (n,p) vary, Proposition 2.1(ii) is suspect.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 3.1, after the isotropic rescaling Y = X/α_{K°}, the comparison estimate (Eq. 21) contains the factor α^{-4}_{K°} multiplying √log(en/m) E|||X|||. To convert this into a body-independent constant C′, the proof invokes Proposition 2.1(ii): for any centrally symmetric K° in B-position with vol_n(K°)=1, the scalar covariance α²_{K°} of the conditioned Gaussian γ_{K°} is bounded below by a universal constant. The entire universal √log(en) bound in Theorems 1.1–1.3 depends on this one external assertion: if α²_{K°} could tend to 0 for some sequence of unit-volume B-position bodies, then the constant in Eq. (21) would blow up and the main theorems would only give a body-dependent (or n-dependent) bound. The manuscript does not prove this proposition, merely cites [10, Prop. 3.11], and Lemma 2.2 (rad_2(K) ≤ c) also hinges on it. The deferred non-symmetric reduction in Section 5.2 is a genuine omission but appears easily fillable via the difference-body argument of Section 6.1; the B-position lower bound is more load-bearing. (Minor issue: Proposition 3.2's statement has '2√T' where the proof uses '2/√T'; Theorem 3.1 uses the correct version.)","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves three universal comparison results in asymptotic convex geometry. For any convex body K in R^n, it shows there exists a linear image TK with the same volume as the crosspolytope such that the Euclidean covering entropy of TK is bounded by a universal constant times that of the crosspolytope (Theorem 1.1). It then proves the sharp order of the minimal spherical-mean-width-to-volume-radius ratio: inf_T M*(TK)/vr(TK) ≤ C sqrt(log(en)), with the crosspolytope and simplex extremal (Theorem 1.2), and extends the bound to all normalized intrinsic volumes/quermassintegrals (Theorem 1.3). The proofs use Eldan's stochastic localization, a comparison inequality for conditioned Gaussian measures in Bobkov's B-position (Theorem 3.1), and a reduction from non-symmetric bodies to difference bodies.","tokens_in":19738,"tokens_out":20540,"duration_ms":173864,"significance":"If correct, the paper resolves the long-standing question of the maximal order of the normalized minimal mean width, without assuming the KLS conjecture. This is a substantial advance over the conditional result of Bizeul–Klartag. The covering-number comparison with the crosspolytope and the quermassintegral bounds are also new and give a satisfying sharp picture. The paper's strengths are its detailed stochastic-localization machinery, the explicit identification of extremal bodies up to universal constants, and the broad scope covering mean width, metric entropy, and all intrinsic volumes. The main caveats are that a key lower bound on the conditioned-Gaussian covariance is imported from Bobkov's work, and that the proof of the headline Theorem 1.2 is deferred.","major_comments":[{"comment":"The proof of Theorem 1.2 is not actually given in Section 5.2: the text says 'We omit the details here' and refers to Section 6.1. Since Theorem 1.2 is the headline result, the non-symmetric reduction via K'=(K-K)/2 should be written out, or the section should explicitly state that the argument of Section 6.1 with k=1 and the volume-radius lower bound (30b) proves the claim. This is fillable from the existing material, but the omission in a main theorem is not acceptable as is.","section":"§5.2"},{"comment":"There are inconsistent constants in the central comparison estimate. Proposition 3.2 as stated has '2√T' but the proof and the later application use '2/√T'; the statement should be corrected. In Eq. (21), substituting T = C_{3.3} α^6 / log(en/m) into Proposition 3.2 yields a factor 2/(α^4 sqrt(C_{3.3})) times sqrt(log(en/m)), not 2 α^{-4} sqrt(C_{3.3}) as displayed. The final constant C' should be adjusted accordingly. These are typos, but they occur in a load-bearing inequality.","section":"§3.3, Eq. (21); Prop. 3.2"},{"comment":"The inequality 'vol_k(P_E T K) ≤ vol_k(P_E T K')' is not literally correct for the difference body. From K - x0 ⊂ 2K' one obtains vol_k(P_E T K) ≤ 2^k vol_k(P_E K''), so the quermassintegral inequality acquires a factor 2: W[k](TK) ≤ 2 W[k](K''). This factor is absorbed by the universal constants, but the displayed estimate should be corrected.","section":"§6.1"},{"comment":"The universality of the constant in Theorem 3.1, and hence of all three main theorems, depends crucially on Bobkov's lower bound α^2_{K^o} ≥ c for unit-volume centrally symmetric bodies in B-position. This proposition is imported from [10] without proof. I do not see circularity, but the dependency is load-bearing. The authors should either state this assumption as a named external theorem with a page/equation reference and a short proof sketch, or give a self-contained proof, since a failure of this bound would make the comparison constant body-dependent.","section":"§2.1, Prop. 2.1(ii)"}],"minor_comments":[{"comment":"The typographical ambiguity in Eq. (21) should be fixed: the factors involving α_{K^o} and sqrt(C_{3.3}) must be typeset unambiguously, and the final expression for C' should be checked against the corrected substitution.","section":"§3.3, Eq. (21)"},{"comment":"The definitions of λ_p^★ and r_p^★ are garbled in the display; they should be λ_p^★ = p / sqrt(log(en/p)) and r_p^★ = sqrt(log(en/p)/p). The subsequent root-taking step should then be written cleanly.","section":"§6.3"},{"comment":"Two nontrivial external results (Lemma 3.7 from [9] and Lemma 6.3 from [25]) are used without proof. Add precise theorem numbers or appendices so a reader can verify the exact statements.","section":"§2.2, Lemma 3.7; §6.3, Lemma 6.3"},{"comment":"The estimates involving m_j = ceil(p/4^j) and the bound p/4^J ≥ 1/(4 rad_2(K)^2) are hard to follow as typeset; please rewrite the chain of inequalities clearly.","section":"§4.1, Eq. (24)-(26)"},{"comment":"There are numerous small OCR/typographical issues (e.g., 'p log(en/p)' instead of 'sqrt(log(en/p))', missing parentheses in display (2), inconsistent use of | || · || |). A careful proofreading pass is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is strong and the central argument appears internally consistent; I found no circularity or fatal error. The main issues are the deferred proof of Theorem 1.2, the constant errors in Proposition 3.2/Eq. (21), and the external dependency on Bobkov's lower bound. If the authors supply the missing details and correct the typos, I would be willing to support acceptance. I do not think rejection is warranted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline result is real: for every convex body K in R^n, after a linear image the spherical mean width is at most C sqrt(log(en)) times the volume radius, with no KLS assumption. The new idea is to run stochastic localization on the conditioned Gaussian measure in Bobkov's B-position rather than on the uniform measure. That switch removes the KLS dependency cleanly, and the covering-number comparison (Theorem 1.1) and the simultaneous quermassintegral bound (Theorem 1.3) are genuinely new. The proof of Theorem 3.1 is the core, and it appears internally consistent; the eigenvalue stopping-time argument with the smoothed f_{beta,m} is laid out in detail and checks out.\n\nThe soft spots are real but not fatal. The proof of Theorem 1.2 is explicitly deferred in Section 5.2, with only a pointer to the difference-body argument used for Theorem 1.3. That argument is sketched rather than fully written, and for the headline theorem the non-symmetric reduction needs more care than one sentence. It is probably fillable, but as it stands the main theorem is not fully verified. Second, the universal constant in Theorem 3.1 depends on Proposition 2.1(ii), imported from Bobkov: a unit-volume centrally symmetric body in B-position has conditioned Gaussian scalar covariance bounded below by a universal constant. The proof divides by alpha^4_{K^o}, so if that lower bound failed, the constant would depend on the body and the sqrt(log n) order would not follow. This is an external result, not derived here, and it is load-bearing. There is also a minor typo in Proposition 3.2: the statement says 2 sqrt(T) but the proof uses 2/sqrt(T), and Theorem 3.1 uses the corrected version.\n\nThe reliance on Mourtada's lemma and on Bobkov's proposition is legitimate if those references are correct, but because the entire paper hinges on the B-position lower bound, I would want a referee to verify that citation carefully. The self-citations to Bizeul-Klartag are appropriate: [9] did only give the bound conditional on KLS. No circularity anywhere.\n\nThis paper is for anyone working in asymptotic convex geometry or high-dimensional probability. It is a serious piece of work with a plausible and likely correct main theorem. The omitted proof and the imported B-position bound should be supplied before publication, but the paper absolutely deserves a full referee rather than a desk rejection.","headline":"Unconditional sqrt(log n) minimal mean width bound that likely resolves the question; main risk is the deferred non-symmetric reduction and reliance on Bobkov's B-position lower bound.","tokens_in":20209,"tokens_out":1736,"would_cite":true,"duration_ms":18058,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A20","52A23","60D05","46B06"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every convex body, a linear image achieves mean width at most sqrt(log n) times its volume radius; the simplex and crosspolytope are extremal.","keywords":["convex body","mean width","Urysohn inequality","covering number","metric entropy","stochastic localization","quermassintegrals","extremal body"],"falsifier":"Find a family of symmetric convex bodies K_n whose polars are in B-position with unit volume but for which the conditioned Gaussian variance parameter alpha^2_{K_n^o} tends to 0 as n grows; or exhibit bodies whose minimal mean-width ratio exceeds C sqrt(log n) for every universal C.","tokens_in":19260,"feed_emoji":"📐","tokens_out":6113,"duration_ms":49216,"temperature":0.7,"pith_summary":"The paper resolves, to sharp order, a classical question in asymptotic convex geometry: among all convex bodies of fixed volume in R^n, which one has the largest possible minimal mean width after an affine rescaling? The answer is the n-simplex and the crosspolytope, up to universal constants. Precisely, for every n and every convex body K, there is an invertible linear map T such that the spherical mean width of TK divided by its volume radius is at most C sqrt(log en), and the order is necessary. The same two bodies also maximize the Euclidean covering entropy and, more generally, all normalized quermassintegrals, up to constants. The proof introduces a comparison inequality for Gaussian measures conditioned on convex bodies and uses stochastic localization, avoiding the need for conditional results based on an isoperimetric conjecture.","feed_headline":"sqrt(log n) bound tames mean width of every convex body","feed_subtitle":"A long-open question in convex geometry is resolved: the simplex and crosspolytope are the extremal shapes.","key_machinery":"The proof rests on two ingredients. First, a position called the B-position (the maximal Gaussian measure position), in which the standard Gaussian conditioned on the polar body has scalar covariance with a variance parameter bounded below by a universal constant. Second, the stochastic localization process, run on this conditioned Gaussian measure; the key comparison inequality (Theorem 3.1) controls the Gaussian expectation of any gauge by the expectation under the conditioned measure plus a Lipschitz term, with a factor sqrt(log(en/m)) that comes from bounding the stopping time at which the m-th smallest covariance eigenvalue drops. Applying the inequality to the support function of K, to","core_discovery":"The central claim is Theorem 1.2: there is a universal constant C such that for every n >= 1 and every convex body K in R^n, 1 <= inf_{T in GL(n)} M*(TK)/vr(TK) <= C sqrt(log(en)), and the upper bound is tight up to universal constants, attained by the crosspolytope B^n_1 and the regular n-simplex. The companion Theorem 1.1 asserts that for any K there is an equal-volume linear image T K whose Euclidean covering number at every scale r is dominated, up to constants, by the covering number of the crosspolytope at a comparable scale. Theorem 1.3 extends the control to all normalized quermassintegrals W[k](TK)/vr(TK), with the same sqrt(log(en/k)) order. These are the first unconditional, sharp","pith_inferences":["The comparison inequality for conditioned Gaussian measures may be a reusable tool for other extremal problems in convex geometry, e.g., bounding other affine-invariant parameters by extremal bodies.","The difference-body reduction for non-symmetric K suggests that the non-symmetric analogues may hold with the same order, though the current proof leaves that step deferred.","A possible testable extension: the same machinery might identify the simplex as the extremal body for the k-th quermassintegral for each fixed k, not just up to constants, in analogy with Ball's reverse isoperimetric theorem."],"forward_implications":["The minimal mean width question is settled unconditionally to sharp order for all convex bodies in all dimensions, independent of the KLS conjecture.","The simplex and crosspolytope are universal extremizers: every convex body's best linear image covers no worse than a constant times the covering numbers of these two bodies at every scale.","The quermassintegral estimates give a simultaneous reverse Alexandrov-type comparison, up to universal constants, for all normalized intrinsic volumes.","The Dvoretzky-number sharpened bound (Theorem 5.1) gives a mean width estimate that adapts to the body's own Euclidean radius."],"fun_headline_variants":["Mean width ratio bounded by sqrt(log n) for all convex bodies","Simplex and crosspolytope extremize convex body mean width","Universal sqrt(log n) bound on convex mean width ratio","Covering numbers maxed by simplex and crosspolytope","Sharp sqrt(log n) bound resolves convex geometry question"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof's universal constant depends on the B-position property that the conditioned Gaussian covariance is scalar and its variance parameter is bounded below by a universal constant; if that lower bound failed for some body, the comparison inequality and hence all three theorems would hold only with a constant depending on the body.","fun_headline_variants_meta":{"raw":{"variants":["Mean width ratio bounded by sqrt(log n) for all convex bodies","Simplex and crosspolytope extremize convex body mean width","Universal sqrt(log n) bound on convex mean width ratio","Covering numbers maxed by simplex and crosspolytope","Sharp sqrt(log n) bound resolves convex geometry question"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000179,"raw_usage":{"total_tokens":1131,"prompt_tokens":731,"completion_tokens":400,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":475,"completion_tokens_details":{"reasoning_tokens":329}},"tokens_in":475,"tokens_out":400,"duration_ms":3943,"temperature":1.0,"reasoning_tokens":329,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T05:19:04.996361+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a family of symmetric convex bodies K_n whose polars are in B-position with unit volume but for which the conditioned Gaussian variance parameter alpha^2_{K_n^o} tends to 0 as n grows; or exhibit bodies whose minimal mean-width ratio exceeds C sqrt(log n) for every universal C.","supporting_citations":[],"review_version":1}