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Moment comparisons, Sudakov inequalities and entropy of centroid bodies

T0 review · 1 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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stated n^{1/4} coefficient: the dyadic sum in (5.5) has an arithmetic slip, and the key Lemma 2.1 is imported from an unpublished preprint. read the letter →

arxiv 2608.10853 v1 pith:HAYF3VSQ submitted 2026-08-11 math.MG math.FAmath.PR

Antonios Hmadi , Dimitris-Marios Liakopoulos This is my paper · ORCID
classification math.MGmath.FAmath.PR MSC 60E1552A2346B0946B07
keywords log-concaverandomvectorsSudakovminorationweakandstrongmomentsL_p-centroidbodiesmetricentropyKLSparametermomentcomparisongauge
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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 $2\leq p\leq n$.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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 $2\leq r\leq n$.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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 $Z_r(X)\subseteq Cr Z_2(X)$.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

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.

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 (1)
  1. [§2, Lemma 2.1; §5, Eqs. (5.3) and (5.5); Theorems 1.2–1.3] 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.
minor comments (5)
  1. [§5, proof of Theorem 1.3, middle-range estimate] 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.
  2. [§6, proof of Lemma 6.2] 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.
  3. [References, [12]] 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.
  4. [§7, proof of Theorem 7.4] 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.
  5. [§3, Theorem 1.1(iii)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; the derivation is self-contained and rests on external first-moment comparisons and unrelated published bounds, not on its own conclusions.

full rationale

I walked the derivation chain from the external first moment comparison (2.4) through Theorems 1.1, 3.2, 4.1, Proposition 4.2, Theorem 1.3, and the entropy results of Sections 6 and 7. The starting point is Bizeul's recorded comparison, whose ingredients (Eldan-Lehec, Bizeul-Klartag, Letwin) are external to this paper. Theorem 1.1 is obtained by applying Gaussian concentration and Poincare's inequality to the gauge; the cube example only shows sharpness. Theorem 3.2 combines (2.4) with Gaussian Sudakov minoration, and the lower bound is an explicit cube computation. The L_p-Sudakov estimates in Section 4 combine (2.4), the classical dual Sudakov inequality (2.3), and an external mean-width estimate for centroid bodies (Lemma 2.1, taken from the unpublished preprint [12]); this is a dependency, not a circular reduction. Theorem 1.3 uses Latała's chaining estimate (5.1) and the scale-dependent SMP constants from (1.4), which are themselves proved from external inputs; the n^{1/4} rate comes from substituting the external bound ell*(Z_q(X)) <= C sqrt(n q ln(e+q)) into the b_q optimization. That bound is not derived from the paper's own conclusions, so no fitted parameter is renamed as a prediction. The entropy and factorization results similarly rely on Mendelson–Milman–Paouris theorems, Latała–Nayar, Paouris' width estimate, and the same external centroid-body width lemma. I found no self-citation that is load-bearing, no uniqueness theorem imported from the authors' prior work, no ansatz smuggled in via citation, and no renaming of a known result. The paper even explicitly distinguishes its Theorem 1.1 from Bizeul's mixed inequality and notes where its estimates are complementary to earlier bounds. The only flagged concern is the unproved, unpublished status of Lemma 2.1, which is a verifiability and dependency issue, not a circularity issue.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

All quantitative conclusions are deductions from a stack of prior results, mostly recent preprints; the paper's own contributions are the moment-order extension and the chaining optimization. No free parameters or invented entities appear. The load-bearing external inputs are listed above.

assumptions (9)
  • domain assumption First moment comparison (2.4): for every gauge φ and every isotropic log-concave X, c/sqrt(ln(en)) Eφ(G) <= Eφ(X) <= C sqrt(ln(en)) Eφ(G).
    Quoted from Bizeul [3, Theorem 1.2], built on Eldan-Lehec, Bizeul-Klartag and Letwin; it is the starting point of Theorem 1.1 and every application.
  • domain assumption Letwin's KLS bound: ψ_n <= C ln^{1/4}(en).
    Used to make the range 1 <= q <= c sqrt(ln(en)) in Theorem 1.1(iii) uniform over X; cited from [22].
  • domain assumption Full-range mean width estimate of Giannopoulos, Pafis and Tziotziou: M*(Z_q(X)) <= C sqrt(q ln(e+q)) for 1 <= q <= n.
    Stated as Lemma 2.1 and taken from the unpublished preprint [12, Equation (3.5)]; it produces the third term in (1.4) and the n^{1/4} coefficient in Theorem 1.3.
  • domain assumption Paouris width estimate: M*(Z_q(X)) <= C sqrt(q) for 1 <= q <= sqrt(n).
    Used in the dyadic sum (5.5) and in Corollaries 6.4 and 6.5 to remove log factors; cited from [26].
  • domain assumption Latała-Nayar theorem (2.8): for every nondegenerate n-dimensional random vector Y and p >= 2, E||Y||_{Z_p(Y)} <= 2 sqrt(e) sqrt((n+p)/p).
    Used in the proof of Theorem 5.1 to bound the mean norm of centroid bodies.
  • domain assumption Latała's SMP lemmas and chaining estimate (5.1).
    Lemma 2.2 items (1)-(5) and the chaining estimate around (5.1) are quoted from [18] and used to assemble the three Sudakov constants and run the dyadic chain.
  • domain assumption MMP regular entropy theorem for ellipsoids [23, Theorem 7.4].
    This is the source of Lemma 6.1, from which the dimension-free packing bound in Theorem 1.4 is derived; it is a published theorem for covariance ellipsoids, not the full MMP conjecture.
  • domain assumption Affirmative solution of the slicing problem: the isotropic constant of any log-concave measure is bounded by an absolute constant.
    Used in Lemma 6.2 for the lower bound I_q(X) >= c sqrt(nq) and in the affine-dimensional estimates of Section 7.
  • standard math Standard background: Gaussian Sudakov minoration, dual Sudakov inequality, Borell's moment comparison, Urysohn and Santaló inequalities.
    Used throughout as classical black boxes; they are stated in Section 2 and cited to standard references.

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Cite this review

Pith. "Pith review of Moment comparisons, Sudakov inequalities and entropy of centroid bodies." pith.science (2026). https://pith.science/paper/HAYF3VSQ

@misc{pith2026260810853,
  author       = {Pith},
  title        = {Pith review of: Moment comparisons, Sudakov inequalities and entropy of centroid bodies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HAYF3VSQ}},
  note         = {Machine review of arXiv:2608.10853}
}
abstract

Let $X$ be an isotropic log-concave random vector in $\mathbb{R}^n$ and let $G$ be standard Gaussian. Starting from the first moment comparison for gauges, we derive corresponding estimates at arbitrary moment orders. For every gauge $\phi$ and every $q\geqslant 1$, $$ \|\phi(G)\|_q\leqslant C\sqrt{\ln(en)+q}\,\|\phi(X)\|_q,\qquad \|\phi(X)\|_q\leqslant C\left(\sqrt{\ln(en)}+\psi(X)\sqrt q\right)\|\phi(G)\|_q. $$ Applied to support functions, this gives the sharp worst case order $C\sqrt{\ln(en)}$ for the $L_2$-Sudakov constant and quantitative $L_p$-Sudakov estimates. Combining, at each level of Latala's dyadic chain, the strongest of the three quantitative $L_p$-Sudakov estimates used here yields $$ \left(\mathbb{E}\|Y\|^p\right)^{1/p}\leqslant C\left(n^{1/4}\sqrt{\ln(en)}\,\ln(e+\ln(en))\,\mathbb{E}\|X\|+\sigma_p(Y)\right) $$ whenever the weak moments of $Y$ are dominated by those of $X$. In a second direction, we study the generalized dual Sudakov problem for the self generated metrics associated with $Z_r(X)^\circ$ and prove dimension free packing estimates for $Z_p(X)$. We also obtain mean norm estimates for centroid bodies, factorization through arbitrary symmetric convex bodies and an affine dimensional refinement.

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Works this paper leans on

30 extracted references · 25 canonical work pages

  1. [12]

    Giannopoulos, M

    A. Giannopoulos, M. Pafis and N. Tziotziou,Geometry of the subgaussian body of an isotropic convex body, Preprint (2026)

    work page 2026

  2. [1]

    Artstein-Avidan, A

    S. Artstein-Avidan, A. Giannopoulos and V. D. Milman,Asymptotic geometric analysis. Part I, Mathematical Surveys and Monographs, 202, American Mathematical Society, Providence, RI, 2015

    work page 2015

  3. [2]

    Artstein-Avidan, A

    S. Artstein-Avidan, A. Giannopoulos and V. D. Milman,Asymptotic geometric analysis. Part II, Mathematical Surveys and Monographs, 261, American Mathematical Society, Providence, RI, 2021

    work page 2021

  4. [3]

    Bizeul,OptimalM M ∗ bounds for convex bodies, Preprint, arXiv:2607.29458 (2026)

    P. Bizeul,OptimalM M ∗ bounds for convex bodies, Preprint, arXiv:2607.29458 (2026)

    arXiv 2026

  5. [4]

    Bizeul and B

    P. Bizeul and B. Klartag,Distances between non-symmetric convex bodies: optimal bounds up to polylog, Preprint, arXiv:2510.20511 (2025)

    arXiv 2025

  6. [5]

    S. G. Bobkov and M. Ledoux,Poincar´ e’s inequalities and Talagrand’s concentration phenomenon for the expo- nential distribution, Probab. Theory Related Fields107(1997), no. 3, 383–400

    work page 1997

  7. [6]

    Borell,Convex measures on locally convex spaces, Ark

    C. Borell,Convex measures on locally convex spaces, Ark. Mat.12(1974), 239–252

    1974

  8. [7]

    Brazitikos, A

    S. Brazitikos, A. Giannopoulos, P. Valettas and B.-H. Vritsiou,Geometry of isotropic convex bodies, Mathe- matical Surveys and Monographs, 196, American Mathematical Society, Providence, RI, 2014

    work page 2014

Show all 30 references
  1. [8]

    Brazitikos and C

    S. Brazitikos and C. Pandis,Geometric bounds for the mean gauge and the mean width in isotropic position, Preprint, arXiv:2608.07216 (2026)

    2026 arXiv

  2. [9]

    Eldan,Thin shell implies spectral gap up to polylog via a stochastic localization scheme, Geom

    R. Eldan,Thin shell implies spectral gap up to polylog via a stochastic localization scheme, Geom. Funct. Anal. 23(2013), no. 2, 532–569. 29

    2013

  3. [10]

    Eldan and J

    R. Eldan and J. Lehec,Bounding the norm of a log-concave vector via thin-shell estimates, in: Geometric Aspects of Functional Analysis, Lecture Notes in Math., 2116, Springer, Cham, 2014, 107–122

    2014

  4. [11]

    Giannopoulos and E

    A. Giannopoulos and E. Milman,M-estimates for isotropic convex bodies and theirL q-centroid bodies, in: Geometric Aspects of Functional Analysis, Lecture Notes in Math., 2116, Springer, Cham, 2014, 159–182

    2014

  5. [13]

    Giannopoulos, G

    A. Giannopoulos, G. Paouris and B.-H. Vritsiou,A remark on the slicing problem, J. Funct. Anal.262(2012), no. 3, 1062–1086

    2012

  6. [14]

    Giannopoulos, P

    A. Giannopoulos, P. Stavrakakis, A. Tsolomitis and B.-H. Vritsiou,Geometry of theL q-centroid bodies of an isotropic log-concave measure, Trans. Amer. Math. Soc.367(2015), no. 7, 4569–4593

    2015

  7. [15]

    Klartag and J

    B. Klartag and J. Lehec,Affirmative resolution of Bourgain ’s slicing problem using Guan ’s bound, Geom. Funct. Anal.35(2025), no. 4, 1147–1168

    2025

  8. [16]

    Lata la,Sudakov minoration principle and supremum of some processes, Geom

    R. Lata la,Sudakov minoration principle and supremum of some processes, Geom. Funct. Anal.7(1997), no. 5, 936–953

    1997

  9. [17]

    Lata la,Weak and strong moments of random vectors, Banach Center Publ.95(2011), 115–121

    R. Lata la,Weak and strong moments of random vectors, Banach Center Publ.95(2011), 115–121

    2011

  10. [18]

    Lata la,Sudakov-type minoration for log-concave vectors, Studia Math.223(2014), no

    R. Lata la,Sudakov-type minoration for log-concave vectors, Studia Math.223(2014), no. 3, 251–274

    2014

  11. [19]

    Lata la,On some problems concerning log-concave random vectors, in: Convexity and Concentration, IMA Vol

    R. Lata la,On some problems concerning log-concave random vectors, in: Convexity and Concentration, IMA Vol. Math. Appl., 161, Springer, New York, 2017, 525–539

    2017

  12. [20]

    Lata la,OnZ p-norms of random vectors, J

    R. Lata la,OnZ p-norms of random vectors, J. Math. Sci. (N.Y.)238(2019), no. 4, 484–494

    2019

  13. [21]

    Lata la and P

    R. Lata la and P. Nayar,Hadamard products and moments of random vectors, Adv. Math.375(2020), 107414

    2020

  14. [22]

    Letwin,The KLS constant isO(log 1/4 n), Preprint, arXiv:2607.24164 (2026)

    B. Letwin,The KLS constant isO(log 1/4 n), Preprint, arXiv:2607.24164 (2026)

    2026 arXiv

  15. [23]

    Mendelson, E

    S. Mendelson, E. Milman and G. Paouris,Generalized dual Sudakov minoration via dimension reduction – a program, Studia Math.244(2019), no. 2, 159–202

    2019

  16. [24]

    Milman,On the mean width of isotropic convex bodies and their associatedL p-centroid bodies, Int

    E. Milman,On the mean width of isotropic convex bodies and their associatedL p-centroid bodies, Int. Math. Res. Not. IMRN (2015), no. 11, 3408–3423

    2015

  17. [25]

    Pajor and N

    A. Pajor and N. Tomczak-Jaegermann,Subspaces of small codimension of finite-dimensional Banach spaces, Proc. Amer. Math. Soc.97(1986), no. 4, 637–642

    1986

  18. [26]

    Paouris,Concentration of mass on convex bodies, Geom

    G. Paouris,Concentration of mass on convex bodies, Geom. Funct. Anal.16(2006), no. 5, 1021–1049

    2006

  19. [27]

    Paouris and R

    G. Paouris and R. Pathak,Optimal mean width and metric entropy estimates for convex bodies, Preprint, arXiv:2607.29522 (2026)

    2026 arXiv

  20. [28]

    Skarmogiannis,On a multi-integral norm defined by weighted sums of log-concave random vectors, Proc

    N. Skarmogiannis,On a multi-integral norm defined by weighted sums of log-concave random vectors, Proc. Amer. Math. Soc.151(2023), no. 10, 4355–4370

    2023

  21. [29]

    V. N. Sudakov,Gaussian measures, Cauchy measures andε-entropy, Soviet Math. Dokl.10(1969), 310–313

    1969

  22. [30]

    Tziotziou,Sections and projections of the outer and inner regularizations of a convex body, Mathematika72 (2026), no

    N. Tziotziou,Sections and projections of the outer and inner regularizations of a convex body, Mathematika72 (2026), no. 3, e70104. Keywords:log-concave random vectors, Sudakov minoration, weak and strong moments,L p-centroid bodies, metric entropy, KLS parameter. 2020 Mathema...

    2026

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