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Strange shadows of $\ell_p$-balls

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Pith's one-line read Random projections of $\ell_p$-balls obey a large deviations principle with an entropy-based rate function over $L_q$-zonoids.

desk verdict Substantial new LDP for random projections of ℓ_p-balls, but the proof of the main rate function has a gap around symmetrization that needs fixing. read the letter →

arxiv 2412.17475 v1 pith:RDMUCA6L submitted 2024-12-23 math.PR math.FA

Zakhar Kabluchko , Mathias Sonnleitner This is my paper · ORCID
classification math.PRmath.FA MSC 52A2352A2152A2260F10
keywords largedeviationsrandomprojectionssectionsL_q-zonoidsℓ_p-ballsmaximumentropyStiefelmanifoldalmostsureconvergence
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 proves a large deviations principle for the $k$-dimensional orthogonal projections of the unit ball of $\ell_p^n$ onto a random subspace, as $n\to\infty$, in the case $2

What carries the argument

The central object is the $L_q$-zonoid: a convex body $K$ whose support function is $h_K(u)=(\int_{\mathbb{R}^k}|\langle x,u\rangle|^q\,d\mu(x))^{1/q}$ for some probability measure $\mu$, so the possible finite-rate shadows are exactly these bodies. The engine of the proof is a large deviations principle for the empirical measures $L_n=\frac1n\sum_{i=1}^n\delta_{\sqrt n v_i}$ formed from the rows $v_i$ of a Haar-distributed Stiefel matrix (Proposition 22); because the map $\mu\mapsto Z_q(\mu)$ is continuous in the appropriate Wasserstein metric, the contraction principle converts that measure-valued LDP into a body-valued LDP with rate $I_p(K)$. The rate function is then re-expressed as a maximum entropy problem, and the small-ball theorem is solved by identifying the maximizer on rotationally invariant classes: for $\beta\le\beta_{k,q}$ the maximizer has density $x\mapsto \omega_k^{-1}Z_{k,q,\beta}^{-1}e^{-\lambda_{k,q,\beta}\|x\|^q}$, and the gap regime would correspond to an exponential density $r^{k-1}e^{-\lambda_1 r^q-\lambda_2 r^2}$ with both parameters positive.

What would settle it

Take $p=\infty$, $k=2$, and $\beta\le\beta_{2,1}$; Theorem 11 gives an explicit constant $c_{2,1,\beta}$. Simulate many independent Haar-distributed projections $\Pi_{n,2}$ for increasing $n$ and compare $n^{-1}\log P[Z_{n,\infty}\subset\beta B_2^2]$ with that constant: a systematic deviation as $n$ grows would refute the large deviations principle, while agreement would confirm the rate function and the continuity argument.

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Extended reading notes

Core claim

The central claim is Theorem 1: for fixed $k\in\mathbb{N}$ and $2<p\le\infty$, the sequence $Z_{n,p}=n^{1/p-1/2}\Pi_{n,k}^\top \mathbb{B}_p^n$ satisfies a large deviations principle in the space of convex bodies equipped with Hausdorff distance, with good rate function $I_p(K)=\inf\{\mathrm{Ent}(\gamma^{\otimes k})-\mathrm{Ent}(\mu):\mu\in\mathcal{P}_q(\mathbb{R}^k),\, Z_q(\mu)=K,\, Z_2(\mu)\subset B_2^k\}$, where $Z_q(\mu)$ is the $L_q$-zonoid generated by $\mu$ and $1/p+1/q=1$. The rate function is finite exactly on $L_q$-zonoids whose generating measure has covariance bounded by the identity, and its unique zero is $m_q B_2^k$, giving the almost sure convergence of Corollary 3. By polarity, the same result transfers to random sections: $K_{n,q}=\Pi_{n,k}^\top(n^{1/q-1/2}\mathbb{B}_q^n\cap E_{n,k})$ has rate function $J_q(K)=I_p(K^\circ)$ (Theorem 5). The proof obtains the body-valued LDP by contracting an LDP for the empirical measure of the rows of the random Stiefel matrix, then rewrites the rate as a maximum entropy problem. A further maximum-entropy computation yields the explicit small-ball exponent in Theorem 11.

Load-bearing premise

The chain of proof imports, without reproof, a large deviations principle for the empirical measures of the rows of a random orthogonal frame; if that imported theorem, or the range of exponents it covers, were not valid, the contraction argument and the subsequent LDP would collapse.

Editorial extensions

If this is right

  • For every fixed $k$ and $2<p\le\infty$, the rescaled projections $Z_{n,p}$ converge almost surely in Hausdorff distance to $m_q B_2^k$; the random sections in the dual range converge almost surely to $m_q^{-1}B_2^k$.
  • Any shadow that appears with merely exponential probability must be an $L_q$-zonoid contained in $B_2^k$ with nonempty interior; all other convex bodies have infinite rate and appear with super-exponentially small probability.
  • The asymptotic probability that $Z_{n,p}\subset\beta B_2^k$ is $\exp(n\,c_{k,q,\beta}+o(n))$ for $\beta\le\beta_{k,q}$, with $c_{k,q,\beta}$ given explicitly in terms of Gamma functions, and is not exponentially small once $\beta\ge m_q$.
  • Continuous functionals such as volume, intrinsic volumes, or diameter inherit a large deviations principle from the rate function by the contraction principle, so their rare deviations can in principle be read off the same entropy functional.
  • In the intermediate range $\beta_{k,q}<\beta<m_q$ the exponential rate exists but is not computed; the paper shows the relevant sets are continuity sets there, so the missing value is well defined.

Reading between the lines

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

  • Beyond the paper: solving the two-moment maximum entropy problem in the gap $\beta_{k,q}<\beta<m_q$ would determine the missing constant $c_{k,q,\beta}$; the expected optimizer is $v(r)=r^{k-1}e^{-\lambda_1 r^q-\lambda_2 r^2}$ with both parameters positive, so the constants can be computed numerically and checked by simulation.
  • Beyond the paper: because every finite-rate shadow is an $L_q$-zonoid with a covariance constraint, extremal questions about volume, mean width, or other functionals of the projection reduce to optimization over a finite-dimensional family of generating measures, which may yield sharper bounds than functional-level large deviations alone.
  • Beyond the paper: the complementary range $1<p<2$ is expected to need speed $n^{2/q}$; if a measure-valued LDP with that speed exists, the same contraction and maximum-entropy machinery would give the analogous body-valued LDP, making the present result a template for the remaining cases.
  • Beyond the paper: the conjectural form of the maximum-entropy density connects the rate function to $q$-stable laws---when the quadratic constraint is dropped, the density is the Fourier transform of a symmetric $q$-stable distribution---so the 'strangeness' of a shadow may be interpretable as a distance in spectral-measure space to Gaussianity.
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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

3 major / 3 minor

Summary. The paper establishes a large deviations principle (LDP) for the sequence of random convex bodies Z_{n,p} = n^{1/p-1/2} Π_{n,k}^T B_p^n, the orthogonal projections of ℓ_p^n balls onto random k-dimensional subspaces, for 2 < p ≤ ∞ (and its dual statement for random sections for 1 ≤ q < 2). The LDP takes place in the space of convex bodies with the Hausdorff metric, and the rate function I_p(K) is expressed as a maximum entropy over probability measures µ generating the L_q-zonoid K, subject to the constraint Z_2(µ) ⊂ B_2^k. The paper derives as corollaries the almost sure convergence of the renormalized projections/sections to a Euclidean ball of radius m_q or m_q^{-1}, computes the exponential rate of the small-ball probability P[Z_{n,p} ⊂ β B_2^k], and proves an auxiliary inequality for the Gamma function. The main proof routes through the Kim–Ramanan LDP for the empirical measures of rows of a Haar-distributed Stiefel matrix, the continuity of the L_q-zonoid map, the contraction principle, and a duality argument for sections.

Significance. If the main theorem is correct, the paper makes a substantial contribution to asymptotic convex geometry and large deviations: it gives a full shape-level LDP for random projections of ℓ_p-balls, with a rate function that is explicit enough to compute small-ball asymptotics, and it unifies several earlier functional and volume results. The approach via L_q-zonoids and maximum entropy is novel and well adapted to the problem. The paper also contains a self-contained proof of an interesting Gamma-function inequality (Proposition 13), and its use of the external Kim–Ramanan LDP is clearly signposted. The main concern is whether the proof of the rate function in Theorem 1 is valid as written; the symmetrization step and equation (2) contain a load-bearing gap that affects all subsequent results.

major comments (3)
  1. [Section 3, proof of Theorem 1 (equations (13)–(14), (2))] The symmetrization step is invalid. For a measure µ with covariance Σ ≤ I and mean m, the symmetrized measure µ̃ = (µ + µ∘(−id))/2 has covariance Σ + mmᵀ, which need not satisfy Σ + mmᵀ ≤ I. If it does not, then H_k(µ̃) = ∞ by the definition in (13), while H_k(µ) < ∞, so the claimed inequality H_k(µ̃) ≤ H_k(µ) fails. A concrete counterexample is µ = N(m, σ²I) with σ²I ≤ I and ||m||² > 1 − σ². Thus the reduction to centered measures, which is used to drop the term (1/2)||bar µ||² from the contraction rate, is not justified. Consequently, the rate function stated in Theorem 1, with the constraint Z_2(µ) ⊂ B_2^k, is not shown to be equal to the contraction rate (14). This gap propagates to Corollary 3, Theorem 5, and Theorem 11, all of which rely on the explicit form of I_p.
  2. [Equation (2) and Remark 2] The statement that Z_2(µ) ⊂ B_2^k is equivalent to Cov(µ) ≤ I_{k×k} is false. By definition Z_2(µ) has support function (∫ |⟨x,u⟩|² dµ(x))^{1/2}, so Z_2(µ) ⊂ B_2^k is equivalent to E[XXᵀ] ≤ I_{k×k}, not to Cov(µ) ≤ I_{k×k}. The two conditions differ by the rank-one matrix mmᵀ, where m = E[X]. They agree only for centered µ. This misstatement is not a minor typo: it is used in the proof of Theorem 1 to replace the covariance constraint from Proposition 22 with the second-moment constraint in the theorem, and it changes the set of admissible measures in the rate function.
  3. [Proposition 20 and Proposition 22] There is an inconsistency between the exact constraint satisfied by the empirical measures L_n and the stated rate function H_k in (13). For every realization of the Stiefel matrix Π_{n,k}, the rows v_i satisfy Σ_{i=1}^n v_i v_iᵀ = ΠᵀΠ = I_k, so the measure L_n = (1/n)Σ δ_{√n v_i} has second moment exactly I_k. The sequence (L_n) therefore lives in the closed set of probability measures with second moment equal to I_k, and any LDP for it must assign infinite rate to measures with second moment different from I_k. However, H_k as defined in (13) is finite for many such measures, e.g. a centered Gaussian with covariance σ²I for σ² < 1. This suggests that either Proposition 22 misstates the Kim–Ramanan result (the correct constraint may be on the second moment rather than the covariance), or the sequence L_n is not the one to which (13) applies. The authors need to reconcile Proposition 22 with the precise statement in [23] and with the identity in Proposition 20.
minor comments (3)
  1. [Throughout] There are a number of typos and OCR artifacts: 'entr opy' in the abstract and Section 1, 'satifies' for 'satisfies', 'δk,q as in Theorem 11 satifies δk,q = k/q a_{k,q}' in Section 5. These should be corrected in a revision.
  2. [Remark 4] The proof sketch for extending the almost sure convergence to 1 < p < ∞ is quite compressed; since this remark is not used later, it is acceptable, but a few more details on the Borel–Cantelli argument would improve readability.
  3. [Section 5, equation (24)] The form of the optimizer v*(r) in (24) is stated without a derivation; the paper refers to [11, Thm. 12.1.1], which is appropriate, but a short explanation of the complementary slackness conditions would make the argument easier to follow.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main LDP is contracted from an external Kim–Ramanan theorem, and the remaining results are derived rather than assumed.

full rationale

The paper's central input is Proposition 22, quoting Kim and Ramanan [23, Thm. 2.8], an external large deviations principle for the empirical measures L_n of the rows of a Haar-distributed Stiefel matrix. This is not a self-citation, and it is independent support with stated hypotheses that do not include the target result. Theorem 1 is obtained from Proposition 22 by the contraction principle together with the continuity of Z_q established in Proposition 15; no fitted quantity is renamed as a prediction. The algebraic rewriting of H_k, the symmetrization step, and the passage between covariance and second-moment constraints are internal derivations, not definitional reductions. Theorem 11 is computed from an explicit maximum-entropy problem, and the Gamma-function inequality in Proposition 13 is proved separately rather than imported. The self-citations that appear, such as [18], [19], and [20], are contextual or concern auxiliary standard facts and are not load-bearing for the main large deviation claim. The skeptic's objection about symmetrization is a potential mathematical correctness issue, not an instance of circularity, since it does not reduce any claim to its own input. The paper is therefore self-contained against its external LDP input, and no circular step is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters fitted to data. The main external assumptions are the Kim-Ramanan Stiefel LDP, the standard contraction principle, and the Cover-Thomas maximum entropy form. No invented entities.

assumptions (4)
  • domain assumption Kim-Ramanan LDP for empirical measures of Haar-distributed Stiefel matrices (Prop. 22, [23, Thm. 2.8])
    Black-box external theorem giving the LDP for L_n = (1/n) Σ δ_{√n v_i}, used to start the contraction argument.
  • standard math Contraction principle for large deviations (Dembo-Zeitouni Thm. 4.2.1)
    Transfers the LDP from measures to convex bodies using the continuous map Z_q: P_q → K_k.
  • standard math Maximum entropy density form exp(-λ1 r^q - λ2 r^2) with complementary slackness (Cover-Thomas Thm. 12.1.1)
    Used in Section 5 to solve the variational problem for the small-ball rate.
  • standard math Borel-Cantelli argument upgrading LDP to almost sure convergence
    Used for Corollaries 3 and 6.

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Pith. "Pith review of Strange shadows of $\ell_p$-balls." pith.science (2026). https://pith.science/paper/RDMUCA6L

@misc{pith2026241217475,
  author       = {Pith},
  title        = {Pith review of: Strange shadows of $\ell_p$-balls},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RDMUCA6L}},
  note         = {Machine review of arXiv:2412.17475}
}
abstract

We prove a large deviations principle for orthogonal projections of the unit ball $\mathbb{B}_p^n$ of $\ell_p^n$ onto a random $k$-dimensional linear subspace of $\mathbb{R}^n$ as $n\to\infty$ in the case $2<p\le \infty$ and for the intersection of $\mathbb{B}_p^n$ with a random $k$-dimensional subspace in the case $1\le p <2$. The corresponding rate function is finite only on $L_q$-zonoids and their duals, respectively, and given in terms of the maximum entropy over suitable measures generating the $L_q$-zonoid, where $\frac{1}{p}+\frac{1}{q}=1$. In particular, we obtain that the renormalized projections/sections almost surely tend to a $k$-dimensional Euclidean ball of certain radius. Moreover, we identify the asymptotic probability that the random orthogonal projection remains within a ball of smaller radius. As a byproduct we obtain an interesting inequality for the Gamma function.

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Works this paper leans on

51 extracted references · 49 canonical work pages

  1. [23]

    S. S. Kim and K. Ramanan. Large deviation principles induced by th e Stiefel manifold, and random multidimensional projections. Electron. J. Probab., 28:Paper No. 169, 23, 2023

    work page 2023

  2. [1]

    Adamczak, P

    R. Adamczak, P. Pivovarov, and P. Simanjuntak. Limit theorems for the volumes of small codimen- sional random sections of ℓn p -balls. Ann. Probab., 52(1):93–126, 2024

    work page 2024

  3. [2]

    Alonso-Guti´ errez, J

    D. Alonso-Guti´ errez, J. Prochno, and C. Th¨ ale. Large deviations for high-dimensional random pro- jections of ℓn p -balls. Adv. in Appl. Math. , 99:1–35, 2018

    work page 2018

  4. [3]

    Ambrosio, N

    L. Ambrosio, N. Gigli, and G. Savar´ e. Gradient flows in metric spaces and in the space of probabilit y measures. Lectures in Mathematics ETH Z¨ urich. Birkh¨ auser Verlag, Basel,2005

    work page 2005

  5. [4]

    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

    work page 2015

  6. [5]

    Aubrun and S

    G. Aubrun and S. J. Szarek. Alice and Bob meet Banach , volume 223 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2017

    work page 2017

  7. [6]

    A. Baci, Z. Kabluchko, J. Prochno, M. Sonnleitner, and C. Th¨ ale . Limit theorems for random points in a simplex. J. Appl. Probab. , 59(3):685–701, 2022

    work page 2022

  8. [7]

    K. Ball. Shadows of convex bodies. Trans. Amer. Math. Soc. , 327(2):891–901, 1991

    work page 1991

Show all 51 references
  1. [8]

    R. Cerf. Large deviations for sums of i.i.d. random compact sets. Proc. Amer. Math. Soc. , 127(8):2431–2436, 1999

    1999

  2. [9]

    Chasapis, P

    G. Chasapis, P. Nayar, and T. Tkocz. Slicing ℓp-balls reloaded: stability, planar sections in ℓ1. Ann. Probab., 50(6):2344–2372, 2022. STRANGE SHADOWS OF ℓp-BALLS 23

    2022

  3. [10]

    J. H. Conway and N. J. A. Sloane. Sphere packings, lattices and groups , volume 290 of Grundlehren der mathematischen Wissenschaften . Springer, New York, third edition, 1999

    1999

  4. [11]

    T. M. Cover and J. A. Thomas. Elements of information theory . Wiley-Interscience, Hoboken, NJ, second edition, 2006

    2006

  5. [12]

    Dembo and O

    A. Dembo and O. Zeitouni. Large deviations techniques and applications , volume 38 of Stochastic Modelling and Applied Probability . Springer-Verlag, Berlin, 2010

    2010

  6. [13]

    L. E. Dor. Potentials and isometric embeddings in L1. Israel J. Math. , 24(3-4):260–268, 1976

    1976

  7. [14]

    W. J. Firey. p-means of convex bodies. Math. Scand. , 10:17–24, 1962

    1962

  8. [15]

    Gantert, S

    N. Gantert, S. S. Kim, and K. Ramanan. Cram´ er’s theorem is at ypical. In Advances in the mathe- matical sciences, volume 6 of Assoc. Women Math. Ser. , pages 253–270. Springer, Cham, 2016

    2016

  9. [16]

    Gantert, S

    N. Gantert, S. S. Kim, and K. Ramanan. Large deviations for ra ndom projections of ℓp balls. Ann. Probab., 45(6B):4419–4476, 2017

    2017

  10. [17]

    S. G. G. Johnston, Z. Kabluchko, and J. Prochno. Projection s of the uniform distribution on the cube: a large deviation perspective. Studia Math. , 264(1):103–119, 2022

    2022

  11. [18]

    Kabluchko and J

    Z. Kabluchko and J. Prochno. Large deviations for random mat rices in the orthogonal group and Stiefel manifold with applications to random projections of product distributions. Ann. Inst. Henri Poincar´ e Probab. Stat., 60(2):990–1024, 2024

    2024

  12. [19]

    Kabluchko, J

    Z. Kabluchko, J. Prochno, and C. Th¨ ale. High-dimensional limit t heorems for random vectors in ℓn p -balls. Commun. Contemp. Math. , 21(1):1750092, 30, 2019

    2019

  13. [20]

    Kabluchko, J

    Z. Kabluchko, J. Prochno, and C. Th¨ ale. A new look at random p rojections of the cube and general product measures. Bernoulli, 27(3):2117–2138, 2021

    2021

  14. [21]

    M. Kanter. The Lp norm of sums of translates of a function. Trans. Amer. Math. Soc. , 79:35–47, 1973

    1973

  15. [22]

    S. S. Kim and K. Ramanan. A conditional limit theorem for high-dime nsional ℓp-spheres. J. Appl. Probab., 55(4):1060–1077, 2018

    2018

  16. [24]

    B. Klartag. A central limit theorem for convex sets. Invent. Math. , 168(1):91–131, 2007

    2007

  17. [25]

    A. L. Koldobskii. Convolution equations in certain Banach spaces . Proc. Amer. Math. Soc. , 111(3):755–765, 1991

    1991

  18. [26]

    Koldobsky

    A. Koldobsky. Fourier analysis in convex geometry , volume 116 of Mathematical Surveys and Mono- graphs. American Mathematical Society, Providence, RI, 2005

    2005

  19. [27]

    Kopp and I

    C. Kopp and I. Molchanov. Large deviations for heavy-tailed ra ndom elements in convex cones. J. Math. Anal. Appl. , 411(1):271–280, 2014

    2014

  20. [28]

    J. Kuelbs. A representation theorem for symmetric stable pro cesses and stable measures on H. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete , 26:259–271, 1973

    1973

  21. [29]

    Liao and K

    Y.-T. Liao and K. Ramanan. Geometric sharp large deviations for random projections of ℓn p spheres and balls. Electron. J. Probab., 29:Paper No. 1, 56, 2024

    2024

  22. [30]

    W. Linde. Moments of measures on Banach spaces. Math. Ann. , 258(3):277–287, 1982

    1982

  23. [31]

    Y. Lonke. Derivatives of the Lp-cosine transform. Adv. Math. , 176(2):175–186, 2003

    2003

  24. [32]

    Lutwak, D

    E. Lutwak, D. Yang, and G. Zhang. Volume inequalities for subsp aces of Lp. J. Differential Geom. , 68(1):159–184, 2004

    2004

  25. [33]

    Matheron

    G. Matheron. Random sets and integral geometry . Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, New York-London-Sydney, 1975

    1975

  26. [34]

    Meyer and A

    M. Meyer and A. Pajor. Sections of the unit ball of ln p . J. Funct. Anal. , 80(1):109–123, 1988. 24 ZAKHAR KABLUCHKO AND MATHIAS SONNLEITNER

    1988

  27. [35]

    Y. Miao. Moderate deviation principles for sums of i.i.d. random com pact sets. Comment. Math. Univ. Carolin., 50(1):103–111, 2009

    2009

  28. [36]

    Mikosch, Z

    T. Mikosch, Z. Pawlas, and G. Samorodnitsky. A large deviation p rinciple for Minkowski sums of heavy-tailed random compact convex sets with finite expectation. J. Appl. Probab. , 48A:133–144, 2011

    2011

  29. [37]

    Molchanov

    I. Molchanov. Theory of random sets . Probability and its Applications. Springer, London, 2005

    2005

  30. [38]

    Molchanov

    I. Molchanov. Convex and star-shaped sets associated with m ultivariate stable distributions. I. Mo- ments and densities. J. Multivariate Anal. , 100(10):2195–2213, 2009

    2009

  31. [39]

    Nayar and T

    P. Nayar and T. Tkocz. Extremal sections and projections of certain convex bodies: a survey. In Harmonic analysis and convexity , volume 9 of Adv. Anal. Geom., pages 343–390. De Gruyter, Berlin, 2023

    2023

  32. [40]

    Paouris and P

    G. Paouris and P. Pivovarov. Small-ball probabilities for the volum e of random convex sets. Discrete Comput. Geom. , 49:601–646, 2013

    2013

  33. [41]

    Paouris, P

    G. Paouris, P. Pivovarov, and J. Zinn. A central limit theorem fo r projections of the cube. Probab. Theory Related Fields , 159(3-4):701–719, 2014

    2014

  34. [42]

    Paouris, P

    G. Paouris, P. Valettas, and J. Zinn. Random version of Dvoret zky’s theorem in ℓn p . Stochastic Process. Appl., 127(10):3187–3227, 2017

    2017

  35. [43]

    J. Prochno. The large and moderate deviations approach in geo metric functional analysis. arXiv e-prints, arXiv:2403.03940 [math.F A], 2024

    2024 arXiv

  36. [44]

    Prochno, C

    J. Prochno, C. Th¨ ale, and P. Tuchel. Limit theorems for the vo lume of random projections and sections of ℓn p -balls. arXiv e-prints , arXiv:2412.16054 [math.PR], 2024

    2024 arXiv

  37. [45]

    Prochno, C

    J. Prochno, C. Th¨ ale, and N. Turchi. Geometry of ℓn p -balls: classical results and recent developments. In High dimensional probability VIII—the Oaxaca volume , volume 74 of Progr. Probab., pages 121–

  38. [46]

    S. T. Rachev and L. R¨ uschendorf. Approximate independenc e of distributions on spheres and their stability properties. Ann. Probab., 19(3):1311–1337, 1991

    1991

  39. [47]

    Schechtman and J

    G. Schechtman and J. Zinn. On the volume of the intersection of two Ln p balls. Proc. Amer. Math. Soc., 110(1):217–224, 1990

    1990

  40. [48]

    Schneider

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

    2014

  41. [49]

    C. Villani. Optimal transport: old and new , volume 338 of A Series of Comprehensive Studies in Mathematics. Springer, 2009

    2009

  42. [50]

    Wendland

    H. Wendland. Scattered Data Approximation . Cambridge University Press, Cambridge, 2004. Zakhar Kabluchko, Institute of Mathematical Stochastics, University of M ¨unster, Orl´ eans-Ring 10, 48149 M ¨unster, Germany Email address : zakhar.kabluchko@uni-muenster.de Mathias Sonnl...

    2004

  43. [150]

    Birkh¨ auser/Springer, Cham, 2019

    2019

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