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, then Prob ({ x ∈ K : ‖x‖2 >… (More)

We introduce complex intersection bodies and show that their properties and applications are similar to those of their real counterparts. In particular, we generalize Busemann’s theorem to the… (More)

We give an alternative proof of a recent result of Klartag on the existence of almost subgaussian linear functionals on convex bodies. If K is a convex body in Rn with volume one and center of mass… (More)

We present proofs of the reverse Santaló inequality, the existence of M ellipsoids and the reverse Brunn–Minkowski inequality, using purely convex geometric tools. Our approach is based on properties… (More)

We study isomorphic properties of two generalizations of intersection bodies the class I k of k-intersection bodies in R and the class BPk of generalized k-intersection bodies in R. In particular, we… (More)

We show that if μ is a centered log-concave probability measure on Rn then, c1 √ n ≤ |Ψ2(μ)| ≤ c2 √ logn √ n , where Ψ2(μ) is the ψ2-body of μ, and c1, c2 > 0 are absolute constants. It follows that… (More)

The purpose of this article is to compare some classical positions of convex bodies. We provide exact quantitative results which show that the minimal surface area position and the minimal mean width… (More)

The purpose of this article is to describe a reduction of the slicing problem to the study of the parameter I1(K,Z ◦ q (K)) = ∫ K ‖〈·, x〉‖Lq(K)dx. We show that an upper bound of the form I1(K,Z ◦ q… (More)

It is known that every isotropic convex bodyK in R has a “subgaussian” direction with constant r = O( √ logn). This follows from the upper bound |Ψ2(K)| 6 c √ logn √ n LK for the volume of the body… (More)