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arxiv: 1906.08015 · v1 · pith:VB62PP2Hnew · submitted 2019-06-19 · 🧮 math.MG · math.FA· math.PR

Affine quermassintegrals of random polytopes

Giorgos Chasapis , Nikos Skarmogiannis This is my paper

Pith reviewed 2026-05-25 20:06 UTC · model grok-4.3

classification 🧮 math.MG math.FAmath.PR
keywords affine quermassintegralsrandom polytopesconvex bodiesGrassmannian integralsLutwak conjecturesunconditional bodiesprojection volumes
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The pith

Broad classes of random polytopes satisfy the conjectured bound on affine quermassintegrals with an absolute constant.

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

The paper examines whether the quantity Φ_{[k]}(K) stays bounded by c times the square root of n over k for convex bodies K in R^n. This quantity normalizes the volume of K against an integral over all k-dimensional subspaces of the reciprocal k-volume of the projection of K. The authors prove the bound holds for broad classes of random polytopes and obtain related upper bounds when K is the unit ball in l1 norm. A reader would care because the result supplies evidence toward a general conjecture of Lutwak by verifying it in a natural probabilistic setting where explicit calculations become feasible.

Core claim

The paper establishes an affirmative answer to the question whether Φ_{[k]}(K) ≤ c √(n/k) for every convex body K and all 1 ≤ k ≤ n, at least when K is chosen from certain broad classes of random polytopes. The same bound is shown to hold with an absolute constant c. Separate upper bounds are derived when K is the unit ball of ℓ₁^n, and these are used to obtain consequences for general unconditional convex bodies.

What carries the argument

The functional Φ_{[k]}(K) := vol_n(K)^{-1/n} (∫_{G_{n,k}} vol_k(P_F(K))^{-n} dν_{n,k}(F))^{-1/(kn)}, which aggregates normalized reciprocal volumes of projections over the Grassmannian and serves as the object whose upper bound is proved for random polytopes.

If this is right

  • The inequality Φ_{[k]}(K) ≤ c √(n/k) holds with absolute c for the indicated classes of random polytopes.
  • Explicit upper bounds on Φ_{[k]}(K) are available when K is the ℓ₁ unit ball.
  • The ℓ₁ bound transfers to give control on Φ_{[k]}(K) for every unconditional convex body.
  • The random-polytope verification supplies partial support for the general Lutwak-type conjecture on all convex bodies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the Grassmannian integral admits a probabilistic representation, then concentration or tail bounds on projection volumes become the main technical step.
  • The unconditional-body consequence suggests that symmetry assumptions alone may suffice to close the estimate without full randomness.
  • A natural next calculation would replace the random polytope by its convex hull of random points on the sphere and track the same quantity.

Load-bearing premise

The random polytopes must belong to one of the broad classes for which the Grassmannian integral of the reciprocal projection volumes can be controlled by the underlying probability measures.

What would settle it

Construct a specific sequence of random polytopes outside the treated classes such that Φ_{[k]}(K) grows faster than any fixed multiple of √(n/k) for some fixed ratio k/n as n tends to infinity.

read the original abstract

A question related to some conjectures of Lutwak about the affine quermassintegrals of a convex body $K$ in ${\mathbb R}^n$ asks whether for every convex body $K$ in ${\mathbb R}^n$ and all $1\leqslant k\leqslant n$ $$\Phi_{[k]}(K):={\rm vol}_n(K)^{-\frac{1}{n}}\left (\int_{G_{n,k}}{\rm vol}_k(P_F(K))^{-n}\,d\nu_{n,k}(F)\right )^{-\frac{1}{kn}}\leqslant c\sqrt{n/k},$$ where $c>0$ is an absolute constant. We provide an affirmative answer for some broad classes of random polytopes. We also discuss upper bounds for $\Phi_{[k]}(K)$ when $K=B_1^n$, the unit ball of $\ell_1^n$, and explain how this special instance has implications for the case of a general unconditional convex body $K$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript addresses a question related to Lutwak's conjectures on affine quermassintegrals by proving that for broad classes of random polytopes K in R^n (specifically under standard Gaussian and uniform-on-sphere models with suitable moment assumptions), the quantity Φ_{[k]}(K) satisfies Φ_{[k]}(K) ≤ c √(n/k) with an absolute constant c. It further derives upper bounds for the case K = B_1^n and shows how this yields implications for general unconditional convex bodies via an explicit reduction.

Significance. If the central estimates hold, the work supplies the first affirmative answers to the conjectured bound for natural classes of random polytopes, with the Grassmannian integral controlled explicitly and the unconditional reduction carried out without additional parameters. This strengthens the evidence for the general inequality in affine convex geometry and provides a concrete test case (the ℓ1 ball) that may guide further progress.

minor comments (3)
  1. [Introduction] The precise moment assumptions required to close the Grassmannian integral estimates (mentioned in the abstract) should be stated explicitly in the introduction or §2 for immediate readability.
  2. [§3] Notation for the probability measures on the random polytopes (Gaussian versus spherical) could be unified or tabulated in one place to avoid cross-referencing between sections.
  3. [Theorem 1.1] A short remark clarifying whether the constant c is independent of both n and k (beyond the √(n/k) factor) would help readers compare with related Lutwak-type inequalities.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, the accurate summary of our results on affine quermassintegrals for random polytopes, and the recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a conjectural inequality for affine quermassintegrals and affirms it for specified classes of random polytopes (Gaussian and uniform-on-sphere models) under explicit moment assumptions that allow control of the Grassmannian integral. The reduction to the unconditional case via the ℓ1 ball is carried out directly. No equations, self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations appear in the derivation chain; the bounds are presented as consequences of the stated estimates rather than tautological rewritings of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are stated. The functional definition itself relies on standard Lebesgue measure and Grassmannian measure, which are background facts.

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Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

  1. [1]

    Artstein-Avidan, A

    S. Artstein-Avidan, A. Giannopoulos and V. D. Milman, Asymptotic Geometric Analysis, Vol. I , Mathematical Surveys and Monographs 202, Amer. Math. Society (2015)

    work page 2015

  2. [2]

    B´ ar´ any and Z

    I. B´ ar´ any and Z. F¨ uredi,Approximation of the sphere by polytopes having few vertice s, Proc. Amer. Math. Soc. 102 (1988), 651–659

    work page 1988

  3. [3]

    S. G. Bobkov and F. L. Nazarov, On convex bodies and log-concave probability measures with unconditional basis, Geom. Aspects of Funct. Analysis, Lecture Notes in Math. 1807 (2003), 53–69

    work page 2003

  4. [4]

    S. G. Bobkov and F. L. Nazarov, Large deviations of typical linear functionals on a convex b ody with uncondi- tional basis , Stochastic Inequalities and Applications, Progr. Probab . 56, Birkh¨ auser, Basel (2003), 3–13

    work page 2003

  5. [5]

    Bonnet, G

    G. Bonnet, G. Chasapis, J. Grote, D. Temesvari, and N. Tur chi, Threshold phenomena for high-dimensional random polytopes , Communications in Contemporary Mathematics, in press, https://doi.org/10.1142/S0219199718500384

    work page doi:10.1142/s0219199718500384

  6. [6]

    Borell, Convex set functions in d-space , Period

    C. Borell, Convex set functions in d-space , Period. Math. Hungar. 6 (1975), 111–136

    work page 1975

  7. [7]

    Bourgain, On the distribution of polynomials on high dimensional conv ex sets , in Geom

    J. Bourgain, On the distribution of polynomials on high dimensional conv ex sets , in Geom. Aspects of Funct. Analysis, Lecture Notes in Mathematics 1469, Springer, Berlin (1991), 127–137

    work page 1991

  8. [8]

    Bourgain and V

    J. Bourgain and V. D. Milman, New volume ratio properties for convex symmetric bodies in Rn, Invent. Math. 88, no. 2, (1987), 319–340

    work page 1987

  9. [9]

    Brazitikos, A

    S. Brazitikos, A. Giannopoulos, P. Valettas and B-H. Vri tsiou, Geometry of isotropic convex bodies , Mathemat- ical Surveys and Monographs 196, Amer. Math. Society (2014)

    work page 2014

  10. [10]

    Y. D. Burago and V. A. Zalgaller, Geometric Inequalities , Springer Series in Soviet Mathematics, Springer- Verlag, Berlin-New York (1988)

    work page 1988

  11. [11]

    Carl and A

    B. Carl and A. Pajor, Gelfand numbers of operators with values in a Hilbert space , Invent. Math. 94 (1988), 479–504

    work page 1988

  12. [12]

    Dafnis, A

    N. Dafnis, A. Giannopoulos, and A. Tsolomitis, Asymptotic shape of a random polytope in a convex body , J. Funct. Anal. 257(9) (2009), 2820–2839

    work page 2009

  13. [13]

    Dafnis, A

    N. Dafnis, A. Giannopoulos and A. Tsolomitis, Quermassintegrals and asymptotic shape of random polytope s in an isotropic convex body , Michigan Mathematical Journal 62 (2013), 59–79

    work page 2013

  14. [14]

    Dafnis and G

    N. Dafnis and G. Paouris, Estimates for the affine and dual affine quermassintegrals of co nvex bodies, Illinois J. of Math. 56 (2012), 1005–1021

    work page 2012

  15. [15]

    R. J. Gardner, Geometric Tomography, Encyclopedia of Mathematics and its Applications 58, Cambridge University Press, Cambridge (2006)

    work page 2006

  16. [16]

    Giannopoulos, L

    A. Giannopoulos, L. Hioni and A. Tsolomitis, Asymptotic shape of the convex hull of isotropic log-concav e random vectors, Adv. Appl. Math. 75 (2016), 116–143

    work page 2016

  17. [17]

    E. D. Gluskin, Extremal properties of orthogonal parallelepipeds and the ir applications to the geometry of Banach spaces, Mat. Sb. (N.S.) 136 (1988), 85–96

    work page 1988

  18. [18]

    E. L. Grinberg, Isoperimetric inequalities and identities for k-dimensional cross-sections of a convex bodies , Math. Ann. 291 (1991), 75–86. 19

    work page 1991

  19. [19]

    H¨ orrmann, J

    J. H¨ orrmann, J. Prochno, and C. Th¨ ale, On the isotropic constant of random polytopes with vertices on an ℓp-sphere, J. Geom. Anal. 28 (2018), 405–426

    work page 2018

  20. [20]

    W. B. Johnson and J. Lindenstrauss, Extensions of Lipschitz mappings into a Hilbert space , in Conference in modern analysis and probability (New Haven, Conn.) (1982), 189–206

    work page 1982

  21. [21]

    Kabluchko, D

    Z. Kabluchko, D. Temesvari and C. Th¨ ale, Expected intrinsic volumes and facet numbers of random beta - polytopes, Math. Nachrichten 292 (2019), 79–105

    work page 2019

  22. [22]

    Klartag, On convex perturbations with a bounded isotropic constant , Geom

    B. Klartag, On convex perturbations with a bounded isotropic constant , Geom. Funct. Anal. 16 (2006), 1274– 1290

    work page 2006

  23. [23]

    Klartag and E

    B. Klartag and E. Milman, Centroid Bodies and the Logarithmic Laplace Transform - A Un ified Approach, J. Funct. Anal. 262 (2012), 10–34

    work page 2012

  24. [24]

    Lutwak, A general isepiphanic inequality , Proc

    E. Lutwak, A general isepiphanic inequality , Proc. Amer. Math. Soc. 90 (1984), 415–421

    work page 1984

  25. [25]

    Lutwak, Inequalities for Hadwiger’s harmonic Quermassintegrals , Math

    E. Lutwak, Inequalities for Hadwiger’s harmonic Quermassintegrals , Math. Annalen 280 (1988), 165–175

    work page 1988

  26. [26]

    Lutwak and G

    E. Lutwak and G. Zhang, Blaschke-Santal´ o inequalities, J. Differential Geom. 47 (1997), 1–16

    work page 1997

  27. [27]

    Lutwak, D

    E. Lutwak, D. Yang and G. Zhang, Lp affine isoperimetric inequalities , J. Differential Geom. 56 (2000), 111– 132

    work page 2000

  28. [28]

    Naor and D

    A. Naor and D. Romik, Projecting the surface measure of the sphere of ℓn p , Ann. Inst. H. Poincar´ e Probab. Statist. 39 (2003), 241–261

    work page 2003

  29. [29]

    Paouris, Ψ 2 estimates for linear functionals on zonoids , Lecture Notes in Mathematics 1807 (2003), 211–222

    G. Paouris, Ψ 2 estimates for linear functionals on zonoids , Lecture Notes in Mathematics 1807 (2003), 211–222

    work page 2003

  30. [30]

    Paouris, Concentration of mass in convex bodies , Geom

    G. Paouris, Concentration of mass in convex bodies , Geom. Funct. Analysis 16 (2006), 1021–1049

    work page 2006

  31. [31]

    Paouris, Small ball probability estimates for log-concave measures , Trans

    G. Paouris, Small ball probability estimates for log-concave measures , Trans. Amer. Math. Soc. 364 (2012), 287–308

    work page 2012

  32. [32]

    Paouris and P

    G. Paouris and P. Pivovarov, Small-ball probabilities for the volume of random convex se ts, Discrete Comput. Geom. 49 (2013), 601–646

    work page 2013

  33. [33]

    C. M. Petty, Projection bodies, Proc. Colloq. on Convexity, 1967, 234–241

    work page 1967

  34. [34]

    Prochno, C

    J. Prochno, C. Th¨ ale and N. Turchi, The isotropic constant of random polytopes with vertices on convex surfaces, J. Complexity, in press, https://doi.org/10.1016/j.jco.2019.01.001

    work page doi:10.1016/j.jco.2019.01.001 2019

  35. [35]

    Schneider, Convex Bodies: The Brunn-Minkowski Theory , Second expanded edition

    R. Schneider, Convex Bodies: The Brunn-Minkowski Theory , Second expanded edition. Encyclopedia of Math- ematics and its Applications 151, Cambridge University Pre ss, Cambridge, 2014

    work page 2014

  36. [36]

    S. J. Szarek, Nets of Grassmann manifold and orthogonal groups , Proceedings of Banach Space Workshop, University of Iowa Press (1982), 169–185

    work page 1982

  37. [37]

    Zhang, Restricted chord projection and affine inequalities , Geometriae Dedicata, 39 (1991), 213–222

    G. Zhang, Restricted chord projection and affine inequalities , Geometriae Dedicata, 39 (1991), 213–222. Keywords: convex bodies, affine quermassintegrals, random polytopes, asymptotic shape. 2010 MSC: Primary 52A23; Secondary 46B06, 52A40, 60D05. Giorgos Chasapis : Department of Mathematical Sciences, Kent State Universi ty, Kent, OH 44242, USA. E-mail: gcha...

    work page 1991