Grigoris Paouris

Small ball probability estimates for log-concave measures

We establish a small ball probability inequality for isotropic log-concave probability measures: there exist absolute constants c1, c2 > 0 such that if X is an isotropic log-concave random vector in R with ψ2 constant bounded by b and if A is a non-zero n × n matrix, then for every e ∈ (0, c1) and y ∈ R, P (‖Ax− y‖2 6 e‖A‖HS) 6 e ( c2 b ‖A‖HS ‖A‖op )2 , where c1, c2 > 0 are absolute constants.

Relative entropy of cone measures and Lp centroid bodies

Let K be a convex body in R n . We introduce a new affine invariant, which we call ΩK ,t hat can be found in three different ways: (a) as a limit of normalized Lp-affine surface areas; (b) as the relative entropy of the cone measure of K and the cone measure of K ◦ ; (c) as the limit of the volume difference of K and Lp-centroid bodies. We investigate properties of ΩK and of related new invariant quantities. In particular, we show new affine isoperimetric inequalities and we show an ‘information inequality’ for convex bodies.

On the singular values of random matrices

We present an approach that allows one to bound the largest and smallest singular values of an N × n random matrix with iid rows, distributed according to a measure on Rn that is supported in a relatively small ball and linear functionals are uniformly bounded in Lp for some p > 8, in a quantitative (non-asymptotic) fashion. Among the outcomes of this approach are optimal estimates of 1 ± c √ n/N not only in the case of the above mentioned measure, but also when the measure is log-concave or when it a product measure of iid random variables with “heavy tails”.

Complex intersection bodies

We introduce complex intersection bodies and show that their properties and applications are similar to those of their real counterparts. In particular, we generalize Busemanns theorem to the complex case by proving that complex intersection bodies of symmetric complex convex bodies are also convex. Other results include stability in the complex Busemann-Petty problem for arbitrary measures and the corresponding hyperplane inequality for measures of complex intersection bodies.

ON THE EXISTENCE OF SUPERGAUSSIAN DIRECTIONS ON CONVEX BODIES

We study the question of whether every centred convex body K of volume 1 in ℝ n has “supergaussian directions”, which means θ ∈ S n −1 such that for all , where c >0 is an absolute constant. We verify that a “random” direction is indeed supergaussian for isotropic convex bodies that satisfy the hyperplane conjecture. On the other hand, we show that if, for all isotropic convex bodies, a random direction is supergaussian then the hyperplane conjecture follows.

A note on subgaussian estimates for linear functionals on convex bodies

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 R n with volume one and center of mass at the origin, there exists x 6 0 such that |{y ∈ K : |h y,xi| > tkh� ,xik 1}| 6 exp(−ct 2 /log 2 (t + 1)) for all t > 1, where c > 0 is an absolute constant. The proof is based on the study of the Lq–centroid bodies of K. Analogous results hold true for general log-concave measures.

Small-Ball Probabilities for the Volume of Random Convex Sets

We prove small-deviation estimates for the volume of random convex sets. The focus is on convex hulls and Minkowski sums of line segments generated by independent random points. The random models considered include (Lebesgue) absolutely continuous probability measures with bounded densities and the class of log-concave measures.

Concentration of mass and central limit properties of isotropic convex bodies

We discuss the following question: Do there exist an absolute constant c > 0 and a sequence Φ(n) tending to infinity with n, such that for every isotropic convex body K in R n and every t ≥ 1 the inequality Prob ({x ∈ K: ∥x∥ 2 ≥ cn√L K t}) ≤ exp ( - Φ(n)t) holds true? Under the additional assumption that K is 1-unconditional, Bobkov and Nazarov have proved that this is true with Φ(n) ≃ √n. The question is related to the central limit properties of isotropic convex bodies. Consider the spherical average f K (t) = ∫ S n-1 |K ∩ (θ⊥ + tθ)|σ(dθ). We prove that for every γ ≥ 1 and every isotropic convex body K in R n , the statements (A) for every t ≥ 1 Prob ({x E K: ∥x∥ 2 ≥ γ√nL K t}) ≤ exp(-Φ(n)t) and (B) for every 0 < t ≤ c 1 (γ)Φ(n)L K , f K (t) ≤ c 2 /L K exp( - t 2 /(c 3 (γ) 2 L 2 K )), where c i (γ) ≃ γ are equivalent.

Random version of Dvoretzky’s theorem in ℓpn

We study the dependence on e in the critical dimension k(n,p,e) for which one can find random sections of the lpn-ball which are (1+e)-spherical. We give lower (and upper) estimates for k(n,p,e) for all eligible values p and e as n→∞, which agree with the sharp estimates for the extreme values p=1 and p=∞. Toward this end, we provide tight bounds for the Gaussian concentration of the lp-norm.

On a local version of the Aleksandrov-Fenchel inequality for the quermassintegrals of a convex body

We discuss the analogue in the Brunn-Minkowski theory of the inequalities of Marcus-Lopes and Bergstrom about symmetric functions of positive reals and determinants of symmetric positive matrices respectively. We obtain a local version of the Aleksandrov-Fenchel inequality W 2 i ≥ W i-1 W i+1 which relates the quermassintegrals of a convex body K to those of an arbitrary hyperplane projection of K. A consequence is the following fact: for any convex body K, for any (n - 1)-dimensional subspace E of R n and any t > 0, |P E (K)+tD E |/|P E (K)| ≤ |K+2tD n |/|K|, where D denotes the Euclidean unit ball and |.| denotes volume in the appropriate dimension.