{"id":"0f75fad0-a8e2-4e51-8f4a-a1652cfea366","arxiv_id":"2608.10853","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New moment comparisons and dyadic chaining yield an n^{1/4} polylog coefficient in Latała's weak-strong inequality for isotropic log-concave vectors, plus sharp L2-Sudakov and polar centroid body entropy estimates.","lead":"The paper proves sharp moment-order comparisons between log-concave random vectors and standard Gaussian vectors, then uses them to improve the best known dimension factor in Latała's weak-strong moment inequality. A specialist reading this will find new Sudakov minoration constants and new packing estimates for centroid bodies.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The n^{1/4} weak–strong coefficient rests on Lemma 2.1, an unproved full-range mean-width bound from unpublished [12]; if [12, Eq. (3.5)] fails or carries extra polylog, the dyadic sum (5.5) and Theorem 1.3 lose the stated rate.","rationale":"The paper's central claim, Theorem 1.3, is a genuine quantitative improvement only if the coefficient n^{1/4} sqrt(ln(en)) ln(e+ln(en)) in (1.7) is attained. The proof's own algebra shows that this coefficient is obtained by summing b_q over dyadic scales, with the upper range q > sqrt(n ln(en)) handled solely by Lemma 2.1. Since Lemma 2.1 is an external unpublished estimate, the main theorem is conditionally correct: it is exactly as strong as that external estimate. The reader identified the same weakest assumption, and I agree. I found no internal inconsistency in the chaining or packing arguments modulo this external input; the sum (5.5) and the net construction are coherent, and the use of (1.6) to transfer d_{X,q}-nets to Y is valid. The paper is also transparent about remaining open factors such as ln(e+ln(en)). The concern is therefore not that the argument is circular or fitted, but that a load-bearing, currently unverifiable external result is necessary for the headline rate. If [12] becomes available and its Eq. (3.5) is correct with the stated polylog, the n^{1/4} claim should stand; if not, the quantitative improvement over the earlier sqrt(n ln(en)) coefficient would have to be re-derived. This does not change the reader's CONDITIONAL verdict.","tokens_in":25916,"tokens_out":10790,"duration_ms":102716,"concrete_test":"Obtain the full manuscript [12] and independently re-derive its Eq. (3.5): prove or disprove ell*(Z_q(X)) <= C sqrt(n q ln(e+q)) for isotropic log-concave X and all 1 <= q <= n, using only published tools (Paouris (2.5), Borell moment comparison, Bizeul's mean-width estimate), and record any extra polylog or range restriction. Then recompute the three dyadic contributions in (5.5): if the true bound is C sqrt(n q) (ln(e+q))^{alpha+1/2} or is only available for q <= sqrt(n), the coefficient in Theorem 1.3 changes from n^{1/4} sqrt(L) ln(e+L) to n^{1/4} sqrt(L) (ln n)^alpha or to sqrt(nL), respectively. This check settles whether (1.7) holds as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.3 hinges on (5.5), the estimate of the dyadic sum of b_q in (5.3). The decisive improvement from sqrt(n ln(en)) to n^{1/4} polylog comes from the third term sqrt(ln(en)) * ell*(Z_q(X)) / q in the range sqrt(n ln(en)) < q <= n. There the paper substitutes Lemma 2.1, which asserts ell*(Z_q(X)) <= C sqrt(n q ln(e+q)) for all 1 <= q <= n, obtaining b_q <= C sqrt(n ln(en) ln(e+q) / q) and hence a geometrically decaying contribution to (5.5). Lemma 2.1 is imported from [12, Eq. (3.5)], an unpublished preprint with no arXiv identifier; the present paper neither proves it nor gives more hypotheses than 'isotropic log-concave.' The same lemma drives the third SMP term in Theorem 1.2(1.4), Corollary 6.4, Corollary 6.5, and the upper bound for I_r used in Theorem 7.4. If the actual full-range bound has an extra (ln(e+q))^alpha with alpha > 0, the upper-range sum in (5.5) acquires a factor (ln(en))^alpha, and the coefficient in (1.7) becomes larger by that polylog; if the valid range is only q <= sqrt(n), or the q-power is worse, the upper range reverts to sqrt(n ln(en))-type terms and Theorem 1.3 is no stronger than earlier results. Thus the central quantitative claim is exactly as secure as an unverifiable external estimate. This is a verifiability and dependency concern, not an accusation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies moment comparisons between an isotropic log-concave vector X in R^n and a standard Gaussian G. It proves that for every gauge φ and every q≥1, ||φ(G)||_q ≤ C√(ln(en)+q) ||φ(X)||_q and ||φ(X)||_q ≤ C(√ln(en)+ψ(X)√q)||φ(G)||_q. These comparisons are applied to support functions to obtain the sharp worst-case L2-Sudakov constant √ln(en), quantitative Lp-Sudakov estimates, and a new weak-strong moment inequality: for Y whose weak moments are dominated by X, (E||Y||^p)^{1/p} ≤ C(n^{1/4}√ln(en) ln(e+ln(en)) E||X|| + σ_p(Y)). A second part develops entropy and packing estimates for polar centroid bodies: Theorem 1.4 gives dimension-free packing for Z_p(X) with respect to the self-generated metric I_r(X) Z_r(X)^°, and Section 7 contains an affine-dimensional refinement and factorization consequences. The paper is clearly written and the internal arguments are mostly self-contained, but a key mean-width lemma (Lemma 2.1) is taken from an unpublished preprint.","tokens_in":26295,"tokens_out":24147,"duration_ms":218679,"significance":"Conditional on Lemma 2.1, the results are significant. The worst-case L2-Sudakov constant is identified up to constants, the weak-strong coefficient improves from √(n ln(en)) to n^{1/4} polylog, and the self-generated packing estimates are dimension-free with explicit dependence on the moment parameter r. The paper also gives a clean dyadic-chaining proof in Section 5 and carefully states the remaining gap to the Mendelson–Milman–Paouris conjecture. The proofs do not appear circular, and Theorem 1.1's moment comparison is of independent interest. The main reservation is that the decisive upper-range estimate in the weak-strong argument relies on an unproved external bound; this is a verifiability concern rather than an internal inconsistency.","major_comments":[{"comment":"Lemma 2.1 asserts ℓ*(Z_q(X)) ≤ C√(nq ln(e+q)) for all 1≤q≤n and is imported from the unpublished preprint [12, Eq. (3.5)] without proof or archive identifier. This bound is the third term in the definition of b_q in (5.3), and the dyadic estimate (5.5) uses it precisely in the range √(n ln(en)) < q ≤ n to obtain the geometrically decaying contribution that yields the n^{1/4}√ln(en) ln(e+ln(en)) coefficient in Theorem 1.3. The same lemma supplies the third SMP term in (1.4), the constants in Corollaries 6.4 and 6.5, and the I_r(X) upper bound used in Theorem 7.4. Because the headline quantitative claims are no stronger than this unverified external estimate, the paper is not self-contained at a load-bearing point. Please prove Lemma 2.1 in the paper, cite a publicly available version of [12], or explicitly state the resulting weaker coefficient in Theorems 1.2 and 1.3.","section":"§2, Lemma 2.1; §5, Eqs. (5.3) and (5.5); Theorems 1.2–1.3"}],"minor_comments":[{"comment":"The sentence 'For √n < q ≤ √n ln(en), Borell's moment comparison and (2.5) give ℓ*(Z_q(X)) ≤ C q n^{1/4}' is terse; please spell out the intermediate step ℓ*(Z_{√n}(X)) ≤ C n^{3/4} from (2.5) and the application of Borell's comparison with r=√n.","section":"§5, proof of Theorem 1.3, middle-range estimate"},{"comment":"The quantity L_X is defined as ||f||_∞^{1/n} only in passing inside the proof; please define it before the display and note that the affirmative slicing result [15] gives L_X ≤ C uniformly for isotropic log-concave vectors.","section":"§6, proof of Lemma 6.2"},{"comment":"Reference [12] is listed as 'Preprint (2026)' with no arXiv identifier; please complete the bibliographic data once the preprint is posted, since the argument depends on it.","section":"References, [12]"},{"comment":"The monotonicity claim for x ↦ x ln(e+A/√x) is stated for A≥0; the displayed derivative is correct for A≥0, but the proof should say explicitly that the inequality continues to hold by continuity when A=0.","section":"§7, proof of Theorem 7.4"},{"comment":"The uniform range q ≤ c√ln(en) in Theorem 1.1(iii) uses Letwin's bound ψ_n ≤ C ln^{1/4}(en) from [22], another unpublished preprint; please state this dependency explicitly in the text.","section":"§3, Theorem 1.1(iii)"}],"recommendation":"major_revision","confidential_remarks":"The single blocking issue is the unproved Lemma 2.1 from the unpublished preprint [12]. If [12] is a companion paper by the same group, it should be posted with a full proof before acceptance. The rest of the argument is readable and, conditional on Lemma 2.1, convincing. I do not see a circularity or an internal inconsistency that would justify rejection, but the paper's main quantitative claims should not be accepted while they rest on an unverifiable external estimate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nTwo things you should know about this paper. First, the genuinely new device is in Theorem 1.3: instead of one global SMP constant, the authors use the best of three Lp-Sudakov bounds at each dyadic level of Latała's chain. That is a good idea and it does yield something better than sqrt(n). Second, the printed proof of that coefficient has a hole. In (5.5) they claim the dyadic sum over sqrt(n ln(en)) < q <= n is O(n^{1/4} sqrt(ln(en))). But substituting their own Lemma 2.1 into b_q gives b_q <= C sqrt(n L ln(e+q)/q) <= C sqrt(n) L / sqrt(q), and the dyadic sum is dominated by the smallest q in the range, producing n^{1/4} L^{3/4} rather than n^{1/4} sqrt(L). So the stated n^{1/4} sqrt(ln(en)) ln(e+ln(en)) coefficient is not established; the argument as written gives an extra (ln(en))^{1/4}. This is independent of the provenance of Lemma 2.1.\n\nThe rest of the paper is in better shape. Theorem 1.1 is a clean extension of the first-moment comparison to all moments, and the sharp order sqrt(ln(en)) for the worst-case L2-Sudakov constant follows immediately. Theorem 1.4's self-generated packing estimate is dimension-free and looks coherent. The paper is transparent about what remains open, and I saw no circularity.\n\nThe real dependency problem is Lemma 2.1, taken from the unpublished preprint [12]. It drives Theorem 1.2's third SMP term, Theorem 1.3, Corollaries 6.4/6.5, and part of Theorem 7.4. The authors state it without proof and give no archive identifier. It might be correct—the claimed bound is plausible—but right now the central quantitative claims rest on an unverifiable input, and on top of that the use of that input in (5.5) is miscalculated.\n\nThis paper deserves a serious referee, but the referee should be asked to verify the arithmetic in Section 5 and to require either a proof of Lemma 2.1 or a reformulation that states the results as conditional on [12]. The audience is specialists in asymptotic geometric analysis; for them the paper raises an interesting structural question even if the headline rate needs adjustment.","headline":"The scale-dependent chaining idea is worth a referee's time, but the proof as written does not support the stated n^{1/4} coefficient: the dyadic sum in (5.5) has an arithmetic slip, and the key Lemma 2.1 is imported from an unpublished preprint.","tokens_in":26863,"tokens_out":16350,"would_cite":false,"duration_ms":143648,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60E15","52A23","46B09","46B07"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper shows that Gaussian-log-concave gauge moment comparisons are nearly tight and yield an $n^{1/4}$ coefficient in the weak-strong bound.","keywords":["log-concave random vectors","Sudakov minoration","weak and strong moments","L_p-centroid bodies","metric entropy","KLS parameter","moment comparison","gauge"],"falsifier":"Compute the mean width $M^*(Z_q(X))$ for an explicit isotropic log-concave vector $X$ with $q$ close to $n$ and compare it with the claimed bound $C\\sqrt{n q \\ln(e+q)}$; if it exceeds that bound by a growing margin, the third SMP term in (1.4) and the $n^{1/4}$ coefficient in Theorem 1.3 are false.","tokens_in":2157,"feed_emoji":"📐","tokens_out":3098,"duration_ms":192140,"temperature":0.7,"pith_summary":"The paper derives moment-order comparisons between Gaussian and log-concave random vectors for gauges, and shows they settle the sharp order of the $L_2$-Sudakov constant and improve the dimensional dependence in the weak-strong problem. For every gauge $\\phi$ and every $q\\geq 1$, the Gaussian $q$-moment of $\\phi$ is controlled by the log-concave $q$-moment times $\\sqrt{\\ln(en)+q}$, and the reverse holds with an added $\\psi(X)\\sqrt{q}$ term. Applied to support functions, this gives $\\sup_X C_X \\simeq \\sqrt{\\ln(en)}$ for the $L_2$-Sudakov constant of isotropic log-concave vectors, and it reduces the weak-strong coefficient from order $\\sqrt{n\\ln(en)}$ to $n^{1/4}\\sqrt{\\ln(en)}\\,\\ln(e+\\ln(en))$. The same machinery gives dimension-free packing estimates for $L_p$-centroid bodies in the self-generated polar metrics, with mean-norm consequences for centroid bodies throughout $2\\leq p\\leq n$.","feed_headline":"Moment comparison nails Sudakov constant at sqrt(log n)","feed_subtitle":"Same argument cuts the weak-strong moment coefficient to n^{1/4} polylog.","key_machinery":"The central object is the $L_p$-centroid body $Z_p(X)$, whose support function is $h_{Z_p(X)}(\\theta)=(\\mathbb{E}|\\langle \\theta,X\\rangle|^p)^{1/p}$; its polar $Z_p(X)^\\circ$ carries the moment metric $d_{X,p}(s,t)=\\|\\langle s-t,X\\rangle\\|_p$. The load-bearing mechanism is the two-sided gauge moment comparison (Theorem 1.1), obtained from the known first-moment comparison via Gaussian concentration and the Poincar\\'e moment inequality. For the weak-strong theorem, the proof feeds three scale-dependent Sudakov-minoration bounds into the dyadic chaining estimate, choosing the strongest coefficient $b_q$ at each dyadic scale; the dyadic sum (5.5) is what produces the $n^{1/4}\\sqrt{\\ln(en)}\\,\\ln(e+\\ln(en))$ factor. For the packing theorem, the inclusion $Z_r(X)\\subseteq Cr Z_2(X)$ and the lower bound $I_r(X)\\geq c\\sqrt{nr}$ turn the regular ellipsoidal entropy estimate into a dimension-free estimate.","core_discovery":"The paper proves two-sided moment comparisons for gauges on isotropic log-concave vectors: $\\|\\phi(G)\\|_q \\leq C\\sqrt{\\ln(en)+q}\\,\\|\\phi(X)\\|_q$ and $\\|\\phi(X)\\|_q \\leq C(\\sqrt{\\ln(en)}+\\psi(X)\\sqrt{q})\\,\\|\\phi(G)\\|_q$. These are promoted from the known first-moment comparison and then applied to support functions, yielding the sharp worst-case $L_2$-Sudakov constant $\\sup_X C_X \\simeq \\sqrt{\\ln(en)}$. The central quantitative result is Theorem 1.3: whenever the weak moments of $Y$ are dominated by those of isotropic log-concave $X$, every norm satisfies $(\\mathbb{E}\\|Y\\|^p)^{1/p} \\leq C(n^{1/4}\\sqrt{\\ln(en)}\\,\\ln(e+\\ln(en))\\,\\mathbb{E}\\|X\\|+\\sigma_p(Y))$, improving the previous dimensional factor from $\\sqrt{n\\ln(en)}$ to $n^{1/4}$ polylog. In the second direction, the paper obtains dimension-free packing estimates for $Z_p(X)$ with respect to the self-generated metrics $Z_r(X)^\\circ$, in particular $\\ln M(Z_p(X), C\\sqrt{r}\\,I_r(X)\\,Z_r(X)^\\circ) \\leq Cp$ for isotropic $X$ and $2\\leq r\\leq n$.","pith_inferences":["The $n^{1/4}$ coefficient is only as trustworthy as the external mean-width bound for $Z_q(X)$ borrowed from an unpublished preprint; a direct check for $q$ near $n$ would quickly indicate whether the third Sudakov term carries real weight.","The factor $\\ln(e+\\ln(en))$ is a bookkeeping term from the intermediate dyadic range; a sharper full-range width estimate would remove it, and the paper leaves open whether it is intrinsic.","The same level-by-level chaining strategy should apply to any family of moment metrics with improving width estimates at intermediate scales, not only centroid bodies.","The self-generated packing theorem reaches the conjectured $e^{Cp}$ bound only after enlarging the separation body by $\\sqrt{r}$; a weaker dependence on $r$ would require a new comparison with the covariance ellipsoid beyond $Z_r(X)\\subseteq Cr Z_2(X)$."],"forward_implications":["The $L_2$-Sudakov constant of every isotropic log-concave vector in $\\mathbb{R}^n$ is at most $C\\sqrt{\\ln(en)}$, and the worst case over such vectors is of exactly this order.","In the weak-strong moment problem, the coefficient in front of $\\mathbb{E}\\|X\\|$ drops from order $\\sqrt{n\\ln(en)}$ to $n^{1/4}\\sqrt{\\ln(en)}\\,\\ln(e+\\ln(en))$, giving the best known dimensional dependence for isotropic log-concave vectors.","For $2\\leq p\\leq n$, the mean norm of the $L_p$-centroid body satisfies $\\sqrt{p}\\,M(Z_p(X))\\leq C\\sqrt{\\ln(en)}$, and the product $M(Z_p(X))M^*(Z_p(X))$ is bounded by $C\\sqrt{\\ln(en)}\\,\\sqrt{\\ln(e+p)}$ in the full range and by $C\\sqrt{\\ln(en)}$ for $p\\leq \\sqrt{n}$.","Packing numbers of $Z_p(X)$ in the self-generated polar metric are dimension free: $\\ln M(Z_p(X), C\\sqrt{r}\\,I_r(X)\\,Z_r(X)^\\circ) \\leq Cp$ for isotropic $X$ and $2\\leq r\\leq n$.","A separated set $T\\subseteq Z_p(X)$ with affine dimension $d$ and separation $aI_r(X)$ in the metric $Z_r(X)^\\circ$ has log-size bounded by $Cp$ plus $d$ times a logarithmic term, giving an affine-dimensional refinement of the entropy estimate."],"supporting_citations":[{"why":"Records the first-moment comparison for gauges from which the paper starts.","marker":"[3]"},{"why":"Gives the reverse comparison from log-concave to Gaussian gauge expectations.","marker":"[4]"},{"why":"Establish the upper comparison of gauge expectations with Gaussian expectations.","marker":"[9,10]"},{"why":"Makes the constants in the first-moment comparison absolute and bounds the Poincar\\'e constant.","marker":"[22]"},{"why":"Supplies Lemma 2.1, the full-range mean-width estimate for centroid bodies on which the third SMP term and Theorem 1.3 depend.","marker":"[12]"},{"why":"Provides the dyadic chaining estimate and the Sudakov-minoration framework used in the weak-strong proof.","marker":"[18]"},{"why":"Connects the $L_2$-Sudakov constant to entropy of polar centroid bodies through $M_p(X)=Z_p(X)^\\circ$.","marker":"[20]"},{"why":"Gives the mean-width bound $M^*(Z_q(X))\\leq C\\sqrt{q}$ used for the low dyadic levels.","marker":"[26]"},{"why":"Supplies the Hadamard-product moment bound used to control mean norms of centroid bodies.","marker":"[21]"},{"why":"Provides the regular entropy estimate for packing centroid bodies against ellipsoids used in Theorem 1.4.","marker":"[23]"}],"fun_headline_variants":["Sudakov constant pinned at sqrt(log n) via moment comparison","Moment bounds yield sharp Sudakov in log n","n^{1/4} factor: new norm bound via Sudakov","Packing estimates for centroid bodies go dimension-free","Sharp L2-Sudakov: sqrt(log n) worst case"],"cache_read_input_tokens":28800,"weakest_assumption_plain":"The load-bearing premise is the mean-width estimate $M^*(Z_q(X)) \\leq C\\sqrt{n q \\ln(e+q)}$ for $1\\leq q\\leq n$, which the paper borrows rather than proves; if this estimate is worse by more than a constant, the third SMP term and the $n^{1/4}$ coefficient in Theorem 1.3 collapse.","fun_headline_variants_meta":{"raw":{"variants":["Sudakov constant pinned at sqrt(log n) via moment comparison","Moment bounds yield sharp Sudakov in log n","n^{1/4} factor: new norm bound via Sudakov","Packing estimates for centroid bodies go dimension-free","Sharp L2-Sudakov: sqrt(log n) worst case"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000179,"raw_usage":{"total_tokens":1398,"prompt_tokens":1139,"completion_tokens":259,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":755,"completion_tokens_details":{"reasoning_tokens":175}},"tokens_in":755,"tokens_out":259,"duration_ms":2949,"temperature":1.0,"reasoning_tokens":175,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:44:59.674840+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the mean width $M^*(Z_q(X))$ for an explicit isotropic log-concave vector $X$ with $q$ close to $n$ and compare it with the claimed bound $C\\sqrt{n q \\ln(e+q)}$; if it exceeds that bound by a growing margin, the third SMP term in (1.4) and the $n^{1/4}$ coefficient in Theorem 1.3 are false.","supporting_citations":[{"cited_title":"Giannopoulos, M","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 2.1, the full-range mean-width estimate for centroid bodies on which the third SMP term and Theorem 1.3 depend."},{"cited_title":"Lata la,Sudakov-type minoration for log-concave vectors, Studia Math.223(2014), no","cited_arxiv_id":null,"evidence_quote":"Provides the dyadic chaining estimate and the Sudakov-minoration framework used in the weak-strong proof."},{"cited_title":"Lata la,OnZ p-norms of random vectors, J","cited_arxiv_id":null,"evidence_quote":"Connects the $L_2$-Sudakov constant to entropy of polar centroid bodies through $M_p(X)=Z_p(X)^\\circ$."},{"cited_title":"Paouris,Concentration of mass on convex bodies, Geom","cited_arxiv_id":null,"evidence_quote":"Gives the mean-width bound $M^*(Z_q(X))\\leq C\\sqrt{q}$ used for the low dyadic levels."},{"cited_title":"Lata la and P","cited_arxiv_id":null,"evidence_quote":"Supplies the Hadamard-product moment bound used to control mean norms of centroid bodies."},{"cited_title":"Mendelson, E","cited_arxiv_id":null,"evidence_quote":"Provides the regular entropy estimate for packing centroid bodies against ellipsoids used in Theorem 1.4."}],"review_version":1}