Poincaré inequality in mean value for Gaussian polytopes

Fleury, Benoît

doi:10.1007/s00440-010-0318-3

Cited by 10 publications

(7 citation statements)

References 45 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For the interesting class of unconditional convex bodies (invariant under reflections with respect to the coordinate hyperplanes), the best known estimate C n = C log(1 + n) was established by B. Klartag [43]. In addition, the conjecture has been established in a certain weak sense for random Gaussian polytopes (with high-probability) by B. Fleury [27].…”

Section: Previously Known Resultsmentioning

confidence: 99%

The KLS isoperimetric conjecture for generalized Orlicz balls

Kolesnikov¹,

Milman²

2018

Ann. Probab.

View full text Add to dashboard Cite

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

show abstract

Section: Previously Known Resultsmentioning

confidence: 99%

The KLS isoperimetric conjecture for generalized Orlicz balls

Kolesnikov¹,

Milman²

2018

Ann. Probab.

View full text Add to dashboard Cite

show abstract

“…Very few positive results are known. It has been proved only for some classes of convex bodies like the unit balls of n p [70,55] and a weaker form is proved for random Gaussian polytopes in [30]. It is also known from the work of Buser [18] and Ledoux [56] that in the case of log-concave probability, the Cheeger constant is related to the best constant in the Poincaré inequality.…”

Section: Almost Extremal Sets In the Isoperimetric Inequalitymentioning

confidence: 99%

Concentration phenomena in high dimensional geometry

Guédon

2014

ESAIM: Proc.

View full text Add to dashboard Cite

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.

show abstract

“…This property is sometimes referred to as isotropicity. It can be also seen by [F,Theorem 3] that the covariance matrix of a typical facet of G α is rather isotropic. On the other hand, in view of the formulas developed in sections 3 and 6, one may expect that the polytope K α exhibits a very different behavior.…”

Section: Comparison With a Gaussian Polytopementioning

confidence: 99%

Volumetric properties of the convex hull of an $n$-dimensional Brownian motion

Eldan

2014

Electron. J. Probab.

View full text Add to dashboard Cite

Let K be the convex hull of the path of a standard Brownian motion B(t) in R n , taken at time 0 ≤ t ≤ 1. We derive formulas for the expected volume and surface area of K. Moreover, we show that in order to approximate K by a discrete version of K, namely by the convex hull of a random walk attained by taking B(tn) at discrete (random) times, the number of steps that one should take in order for the volume of the difference to be relatively small is of order n 3 . Next, we show that the distribution of facets of K is in some sense scale invariant: for any given family of simplices (satisfying some compactness condition), one expects to find in this family a constant number of facets of tK as t → ∞. Finally, we discuss some possible extensions of our methods and suggest some further research.

show abstract

Poincaré inequality in mean value for Gaussian polytopes

Abstract: Let K N =[±G 1 , . . . , ±G N ] be the absolute convex hull of N independent standard Gaussian random points in R n with N ≥ n. We prove that, for any 1-Lipschitz function f : R n → R, the polytope K N satisfies the following Poincaré inequality in mean value:

Cited by 10 publications

References 45 publications

The KLS isoperimetric conjecture for generalized Orlicz balls

The KLS isoperimetric conjecture for generalized Orlicz balls

Concentration phenomena in high dimensional geometry

Volumetric properties of the convex hull of an $n$-dimensional Brownian motion

Contact Info

Product

Resources

About