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REVIEW 4 major objections 5 minor 1 cited by

Optimal mean width and metric entropy estimates for convex bodies

T0 review · 4 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2607.29522 v1 pith:YRI77OHM submitted 2026-07-31 math.MG math.FAmath.PR

Grigoris Paouris , Reese Pathak This is my paper · ORCID
classification math.MGmath.FAmath.PR MSC 52A2052A2360D0546B06
keywords convexbodymeanwidthUrysohninequalitycoveringnumbermetricentropystochasticlocalizationquermassintegralsextremal
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 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.

What carries the argument

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

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

4 major / 5 minor

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.

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 (4)
  1. [§5.2] 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.
  2. [§3.3, Eq. (21); Prop. 3.2] 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.
  3. [§6.1] 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.
  4. [§2.1, Prop. 2.1(ii)] 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.
minor comments (5)
  1. [§3.3, Eq. (21)] 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.
  2. [§6.3] 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.
  3. [§2.2, Lemma 3.7; §6.3, Lemma 6.3] 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.
  4. [§4.1, Eq. (24)-(26)] 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.
  5. [Throughout] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation found: Theorem 1.2 follows from a stochastic-localization comparison theorem and Bobkov's external B-position properties; the extremal ratios are checked against known crosspolytope/simplex entropy benchmarks rather than assumed.

full rationale

The paper's central claim (Theorem 1.2, Eq. 3) is not obtained by assuming the target bound. Its engine is Theorem 3.1, a comparison inequality between Gaussian expectations under γ_n and a conditioned Gaussian measure γ_{K°}; that inequality is proved from Eldan's stochastic localization, Propositions 3.2–3.3 and Lemmas 3.4–3.8, without invoking the desired √log(en) bound. The universal constant in Eq. (21) does depend on Proposition 2.1(ii) (α²_{K°} ≥ c), imported from Bobkov [10]; this is an external theorem, not an input of the target result, so the dependence is fragility, not circularity. The crosspolytope and simplex appear only as comparison benchmarks: Schuett's entropy formula (Eq. 2) and known intrinsic-volume estimates supply the right-hand sides, and the proof shows every B-position body's entropy is controlled by those fixed benchmark quantities (Corollary 4.4). The self-citations [21] and [26] are contextual remarks about positions and known cases, not load-bearing steps. The only flagged omission is the deferred non-symmetric reduction in Section 5.2 (“We omit the details here”), but it explicitly points to the same difference-body argument used in Section 6.1 and relies on standard Rogers–Shephard/Brunn–Minkowski/Santaló inequalities, not on the conclusion. The Proposition 3.2 display containing “2√T” is a typo (the proof and Theorem 3.1 use 2/√T) and is a correctness consistency issue, not a circularity. No equation is shown to reduce to its own input, and no fitted parameter is relabeled as a prediction.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No data-fitted parameters: all constants are universal existential constants chosen in the proof. The paper introduces no new physical or geometric entities; B-position and conditioned Gaussians are existing tools. The proof is parameter-free in the sense that no empirical constants are fitted, though several imported theorems are used as axioms.

assumptions (7)
  • domain assumption Bobkov's B-position properties: scalar covariance alpha^2 I and alpha^2 >= c when vol(K^o)=1
    Imported in Section 2.1 from [10]; used in Section 3.3 Eq. (21) to keep constants universal.
  • standard math Stochastic localization SDE and Ito calculus for f_{beta,m}(A_t)
    Used throughout Section 3; standard in the Eldan-Lehec/Bizeul-Klartag framework, cited to [13,14,15,9].
  • standard math Schuett's crosspolytope entropy estimates (2)
    External benchmark in Corollary 4.4; the target is a comparison, not a derivation of (2).
  • standard math Reverse Blaschke-Santalo and Rogers-Shephard inequalities
    Used in Sections 4.2, 6.1, and Lemma 6.4 to transfer volume normalizations between K and its difference body/polar.
  • standard math Sudakov minoration and Levy concentration on the sphere
    Used in Lemma 4.2 and Theorem 6.1.
  • domain assumption Mourtada's Lemma 6.3 linking intrinsic volumes to entropy and width
    Imported from [25]; load-bearing for Proposition 6.2.
  • domain assumption Existence of a linear map placing K^o in B-position with unit volume
    Used in Sections 4.2, 5.2, 6.1; follows from Bobkov/Milman M-position theory, not proved in the paper.

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Pith. "Pith review of Optimal mean width and metric entropy estimates for convex bodies." pith.science (2026). https://pith.science/paper/YRI77OHM

@misc{pith2026260729522,
  author       = {Pith},
  title        = {Pith review of: Optimal mean width and metric entropy estimates for convex bodies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YRI77OHM}},
  note         = {Machine review of arXiv:2607.29522}
}
abstract

We show that there exists a constant $C > 0$ such that for any $n \geq 1$ and any convex body $K \subset \mathbf{R}^n$, \[ 1 \leq \inf_{T \in \mathrm{GL}(n)} \, \frac{M^\ast(TK)}{\mathrm{vr}(TK)} \leq C\sqrt{\log(\mathrm{e} n)}, \] where $M^\ast$ denotes the spherical mean width and $\mathrm{vr}(\cdot)$ denotes the volume radius. The righthand side is attained, up to universal constants, by the crosspolytope and the regular $n$-simplex. Analogously, we show that, up to universal constants, the logarithm of the Euclidean covering number is maximized over convex bodies $K \subset \mathbf{R}^n$ by the simplex and crosspolytope. Our proof makes use of Eldan's stochastic localization.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Geometric Bounds for the Mean Gauge and the Mean Width in Isotropic Position

    math.MG 2026-08 accept novelty 6.0 of 10

    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.

Reference graph

Works this paper leans on

39 extracted references · cited by 1 Pith paper

  1. [9]

    Bizeul and B

    P. Bizeul and B. Klartag. Distances between non-symmetric convex bodies: optimal bounds up to polylog, 2026

    2026

  2. [10]

    S. G. Bobkov. On Milman’s ellipsoids and M -position of convex bodies. In Concentration, functional inequalities and isoperimetry, volume 545 of Contemp. Math. , pages 23–33. Amer. Math. Soc., Providence, RI, 2011

    2011

  3. [1]

    Artstein, V

    S. Artstein, V. Milman, and S. J. Szarek. Duality of metric entropy. Ann. of Math. (2) , 159(3):1313–1328, 2004

    2004

  4. [2]

    Artstein-Avidan, A

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

    2015

  5. [3]

    Aubrun and S

    G. Aubrun and S. a. J. Szarek. Alice and Bob meet Banach , volume 223 of Mathematical Surveys and Mono- graphs. American Mathematical Society, Providence, RI, 2017. The interface of asymptotic geometric analysis and quantum information theory

    2017

  6. [4]

    K. Ball. Volume ratios and a reverse isoperimetric inequality. J. London Math. Soc. (2) , 44(2):351–359, 1991

    1991

  7. [5]

    K. M. Ball. Isometric problems in lp and sections of convex sets . PhD thesis, University of Cambridge, 1986

    1986

  8. [6]

    F. Barthe. On a reverse form of the Brascamp-Lieb inequality. Invent. Math. , 134(2):335–361, 1998

    1998

Show all 39 references
  1. [7]

    Barthe and D

    F. Barthe and D. Cordero-Erausquin. Invariances in variance estimates.Proc. Lond. Math. Soc. (3), 106(1):33–64, 2013

    2013

  2. [8]

    P. Bizeul. On measures strongly log-concave on a subspace. Ann. Inst. Henri Poincar´ e Probab. Stat., 60(2):1090– 1100, 2024

    2024

  3. [11]

    Bourgain and V

    J. Bourgain and V. D. Milman. New volume ratio properties for convex symmetric bodies in Rn. Invent. Math., 88(2):319–340, 1987

    1987

  4. [12]

    Carl and I

    B. Carl and I. Stephani. Entropy, compactness and the approximation of operators , volume 98 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1990. 20 GRIGORIS PAOURIS AND REESE PATHAK

    1990

  5. [13]

    R. Eldan. Distribution of Mass in Convex Bodies . PhD thesis, Tel Aviv University, 2012

    2012

  6. [14]

    R. Eldan. Thin shell implies spectral gap up to polylog via a stochastic localization scheme. Geom. Funct. Anal., 23(2):532–569, 2013

    2013

  7. [15]

    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 , volume 2116 of Lecture Notes in Math. , pages 107–122. Springer, Cham, 2014

    2014

  8. [16]

    Figiel and N

    T. Figiel and N. Tomczak-Jaegermann. Projections onto Hilbertian subspaces of Banach spaces. Israel J. Math. , 33(2):155–171, 1979

    1979

  9. [17]

    A. A. Giannopoulos and V. D. Milman. Extremal problems and isotropic positions of convex bodies. Israel J. Math., 117:29–60, 2000

    2000

  10. [18]

    A. A. Giannopoulos, V. D. Milman, and M. Rudelson. Convex bodies with minimal mean width. In Geometric aspects of functional analysis , volume 1745 of Lecture Notes in Math. , pages 81–93. Springer, Berlin, 2000

    2000

  11. [19]

    Klartag and J

    B. Klartag and J. Lehec. Bourgain’s slicing problem and KLS isoperimetry up to polylog. Geom. Funct. Anal., 32(5):1134–1159, 2022

    2022

  12. [20]

    Y. T. Lee and S. S. Vempala. Eldan’s stochastic localization and the KLS conjecture: isoperimetry, concentration and mixing. Ann. of Math. (2) , 199(3):1043–1092, 2024

    2024

  13. [21]

    Markessinis, G

    E. Markessinis, G. Paouris, and C. Saroglou. Comparing the M -position with some classical positions of convex bodies. Math. Proc. Cambridge Philos. Soc. , 152(1):131–152, 2012

    2012

  14. [22]

    B. Maurey. Type, cotype and K-convexity. In Handbook of the geometry of Banach spaces, Vol. 2 , pages 1299–

  15. [23]

    V. D. Milman. In´ egalit´ e de Brunn-Minkowski inverse et applications ` a la th´ eorie locale des espaces norm´ es.C. R. Acad. Sci. Paris S´ er. I Math., 302(1):25–28, 1986

    1986

  16. [24]

    V. D. Milman and A. Pajor. Entropy and asymptotic geometry of non-symmetric convex bodies. Adv. Math. , 152(2):314–335, 2000

    2000

  17. [25]

    Mourtada

    J. Mourtada. Universal coding, intrinsic volumes, and metric complexity. To appear in the Journal of the Euro- pean Mathematical Society, 2025

    2025

  18. [26]

    Paouris and P

    G. Paouris and P. Valettas. On Dvoretzky’s theorem for subspaces of Lp. J. Funct. Anal. , 275(8):2225–2252, 2018

    2018

  19. [27]

    A. Pinkus. n-widths in approximation theory , volume 7 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] . Springer-Verlag, Berlin, 1985

    1985

  20. [28]

    G. Pisier. Remarques sur un r´ esultat non publi´ e de B. Maurey. InSeminar on Functional Analysis, 1980–1981 , pages Exp. No. V, 13. ´Ecole Polytech., Palaiseau, 1981

    1980

  21. [29]

    G. Pisier. Holomorphic semigroups and the geometry of Banach spaces. Ann. of Math. (2) , 115(2):375–392, 1982

    1982

  22. [30]

    G. Pisier. The volume of convex bodies and Banach space geometry , volume 94 of Cambridge Tracts in Mathe- matics. Cambridge University Press, Cambridge, 1989

    1989

  23. [31]

    O. Regev. Some questions related to the reverse Minkowski theorem. In ICM—International Congress of Math- ematicians. Vol. 6. Sections 12–14 , pages 4898–4912. EMS Press, Berlin, [2023] ©2023

    2023

  24. [32]

    Schneider

    R. Schneider. Convex bodies: the Brunn-Minkowski theory , volume 151 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, expanded edition, 2014

    2014

  25. [33]

    Sch¨ utt

    C. Sch¨ utt. Entropy numbers of diagonal operators between symmetric Banach spaces. J. Approx. Theory , 40(2):121–128, 1984

    1984

  26. [34]

    G. R. Shorack and J. A. Wellner. Empirical processes with applications to statistics , volume 59 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009. Reprint of the 1986 original [MR0838963]

    2009

  27. [35]

    Talagrand

    M. Talagrand. Regularity of Gaussian processes. Acta Math., 159(1-2):99–149, 1987

    1987

  28. [36]

    Talagrand

    M. Talagrand. The generic chaining. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2005. Upper and lower bounds of stochastic processes

    2005

  29. [37]

    Vershynin

    R. Vershynin. High-dimensional probability, volume 47 of Cambridge Series in Statistical and Probabilistic Math- ematics. Cambridge University Press, Cambridge, 2018. An introduction with applications in data science, With a foreword by Sara van de Geer

    2018

  30. [38]

    Vritsiou

    B.-H. Vritsiou. Regular ellipsoids and a Blaschke-Santal´ o-type inequality for projections of non-symmetric convex bodies. J. Funct. Anal. , 286(11):Paper No. 110414, 40, 2024. Department of Mathematics, Texas A&M University, and Department of Mathematics, Princeton Universit...

    2024

  31. [1332]

    North-Holland, Amsterdam, 2003

    2003

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