Concentration in a thin Euclidean shell for log-concave measures

Fleury, Benoît

doi:10.1016/j.jfa.2010.04.019

Cited by 35 publications

(39 citation statements)

References 11 publications

(11 reference statements)

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“…More than two decades after being put forth, the KLS conjecture is still unresolved, and the presently best known (dimension-dependent) estimate on C = C n in (1.4) is C n ≤ Cn 1/4 , obtained very recently (after this work was posted on the arXiv) by Y. T. Lee and S. Vempala [52] by employing the remarkable Stochastic Localization method of R. Eldan [23]; previous contributions include those by KLS [37], S. Bobkov [11], B. Klartag [40,41], B. Fleury [26] and O. Guédon and Milman [34]. The conjecture has been confirmed (uniformly in n) for unit-balls of ℓ n p (by S. Sodin [67] when p ∈ [1, 2] and R. Lata la and J. Wojtaszczyk [49] when p ∈ [2, ∞]), the simplex by F. Barthe and P. Wolff [7], convex bodies of revolution by N. Huet [36], convex sets of bounded volume-ratio constructed in a certain manner from log-concave measures which satisfy the conjecture [46], linear images and Cartesian products of these subclasses (see for the latter) and various perturbations thereof [56,59].…”

Section: Previously Known Resultsmentioning

confidence: 99%

The KLS isoperimetric conjecture for generalized Orlicz balls

Kolesnikov¹,

Milman²

2018

Ann. Probab.

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What is the optimal way to cut a convex bounded domain K in Euclidean space (R n , |·|) into two halves of equal volume, so that the interface between the two halves has least surface area? A conjecture of Kannan, Lovász and Simonovits asserts that, if one does not mind gaining a universal numerical factor (independent of n) in the surface area, one might as well dissect K using a hyperplane. This conjectured essential equivalence between the former nonlinear isoperimetric inequality and its latter linear relaxation, has been shown over the last two decades to be of fundamental importance to the understanding of volume-concentration and spectral properties of convex domains. In this work, we address the conjecture for the subclass of generalized Orlicz balls

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Section: Previously Known Resultsmentioning

confidence: 99%

The KLS isoperimetric conjecture for generalized Orlicz balls

Kolesnikov¹,

Milman²

2018

Ann. Probab.

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“…Therefore, to prove the lower bound for 0 < p < ∞, it is sufficient to consider K in isotropic position. Let L K be the isotropic constant of K. By the thin shell estimate of O. Guédon and E.Milman [22] (see also [15,42]), we have with universal constants c and C, that for all t ≥ 0,…”

Section: Proofsmentioning

confidence: 98%

Constrained convex bodies with extremal affine surface areas

Giladi

Huang

Schütt

et al. 2020

Journal of Functional Analysis

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Given a convex body K ⊆ R n and p ∈ R, we introduce and study the extremal inner and outer affine surface areaswhere asp(K ′ ) denotes the Lp-affine surface area of K ′ , and the supremum is taken over all convex subsets of K and the infimum over all convex compact subsets containing K. The convex body that realizes IS1(K) in dimension 2 was determined in [3] where it was also shown that this body is the limit shape of lattice polytopes in K. In higher dimensions no results are known about the extremal bodies. We use a thin shell estimate of [22] and the Löwner ellipsoid to give asymptotic estimates on the size of ISp(K) and osp (K). Surprisingly, it turns out that both quantities are proportional to a power of volume. *

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“…In a breakthrough, Eldan [35] showed that the thin shell conjecture is in fact equivalent to the KLS conjecture up a logarithmic factor (see Theorem 35). [52] n 4/10 2010/Fleury [39] n 3/8 2011/Guedon-Milman [44] n 1/3 2016/Lee-Vempala [65] n 1/4 Table 1: Progress on the thin shell bound.…”

Section: Thin Shell To Central Limit Theoremmentioning

confidence: 99%

The Kannan–Lovász–Simonovits conjecture

Lee

Vempala

2017

Current Developments in Mathematics

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The Kannan-Lovász-Simonovits conjecture says that the Cheeger constant of any logconcave density is achieved to within a universal, dimension-independent constant factor by a hyperplane-induced subset.Here we survey the origin and consequences of the conjecture (in geometry, probability, information theory and algorithms) as well as recent progress resulting in the current best bounds. The conjecture has lead to several techniques of general interest. * This is a working draft. We will be grateful for any comments!

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Concentration in a thin Euclidean shell for log-concave measures

Cited by 35 publications

References 11 publications

The KLS isoperimetric conjecture for generalized Orlicz balls

The KLS isoperimetric conjecture for generalized Orlicz balls

Constrained convex bodies with extremal affine surface areas

The Kannan–Lovász–Simonovits conjecture

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