On Isoperimetric Constants for Log-Concave Probability Distributions

Bobkov, Sergey G.

doi:10.1007/978-3-540-72053-9_4

Cited by 32 publications

(23 citation statements)

References 34 publications

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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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“…After these pioneer works, many developments appear to unify and to generalize these observations [7,9,10,6]. We refer to [34] for a detailed bibliography.…”

Section: From Concentration To Functional Inequalitiesmentioning

confidence: 94%

From concentration to logarithmic Sobolev and Poincaré inequalities

Gozlan¹,

Roberto²,

Samson³

2011

Journal of Functional Analysis

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We give a new proof of the fact that Gaussian concentration implies the logarithmic Sobolev inequality when the curvature is bounded from below, and also that exponential concentration implies Poincaré inequality under null curvature condition. Our proof holds on non-smooth structures, such as length spaces, and provides a universal control of the constants. We also give a new proof of the equivalence between dimension free Gaussian concentration and Talagrand's transport inequality.

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“…The localization method was further developed in [32,33]. And Bobkov [10] improved this result to h ≥ c…”

Section: Conjecture 2 (The Kls Conjecture)mentioning

confidence: 99%

Concentration phenomena in high dimensional geometry

Guédon

2014

ESAIM: Proc.

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The purpose of this note is to present several aspects of concentration phenomena in high dimensional geometry. At the heart of the study is a geometric analysis point of view coming from the theory of high dimensional convex bodies. The topic has a broad audience going from algorithmic convex geometry to random matrices. We have tried to emphasize different problems relating these areas of research. Another connected area is the study of probability in Banach spaces where some concentration phenomena are related with good comparisons between the weak and the strong moments of a random vector.

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On Isoperimetric Constants for Log-Concave Probability Distributions

Cited by 32 publications

References 34 publications

The KLS isoperimetric conjecture for generalized Orlicz balls

The KLS isoperimetric conjecture for generalized Orlicz balls

From concentration to logarithmic Sobolev and Poincaré inequalities

Concentration phenomena in high dimensional geometry

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