Concentration of mass on convex bodies

Abstract

We establish a sharp concentration of mass inequality for isotropic convex bodies: there exists an absolute constant c > 0 such that if K is an isotropic convex body in R n , thenfor every t 1, where LK denotes the isotropic constant.

Upper bounds for the volume and diameter of $m$-dimensional sections of convex bodies

“…It is well known that this fact is equivalent to the slicing problem: there exists an absolute constant c > 0 with the following property: if K is a convex body in R n there exists a hyperplane H such that |K ∩ H| n−1 > c. The best known bound for the isotropy constant was given by Bourgain L K ≤ Cn 1/4 log n (see [Bo1], [Bo2] and also [D] and [Pa1] for the nonsymmetric case). Very recently, using a new concentration inequality of G. Paouris ([Pa2], [Pa3]), B. Klartag has just improved Bourgain's estimate proving that L K ≤ Cn 1/4 (see [Kl2]). Let us assume that the convex body K of volume one has its centroid in the origin.…”

Quantum ergodicity of random orthonormal bases of spaces of high dimension

“…Note that the concentration of mass of these convex bodies in high-dimensional spaces is evidently quite different from that of isotropic convex bodies [6].…”

ℓ 1-regularized linear regression: persistence and oracle inequalities

“…As we showed above, in this case Q L 2 ≤ c log nd ≤ ch 2 1 . To bound M , we need the following deep result of Paouris [25]: Theorem 4.10 There are absolute constants c 1 and c 2 for which the following holds. Let X be distributed according to an isotropic log-concave measure on…”