Petros Valettas

Geometry of isotropic convex bodies

Background from asymptotic convex geometry Isotropic log-concave measures Hyperplane conjecture and Bourgains upper bound Partial answers L q -centroid bodies and concentration of mass Bodies with maximal isotropic constant Logarithmic Laplace transform and the isomorphic slicing problem Tail estimates for linear functionals M and M? *-estimates Approximating the covariance matrix Random polytopes in isotropic convex bodies Central limit problem and the thin shell conjecture The thin shell estimate Kannan-Lov sz-Simonovits conjecture Infimum convolution inequalities and concentration Information theory and the hyperplane conjecture Bibliography Subject index Author index

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 the Geometry of Log-Concave Probability Measures with Bounded Log-Sobolev Constant

Let \(\mathcal{L}S_{lc}(\kappa )\) denote the class of log-concave probability measures μ on \({\mathbb{R}}^{n}\) which satisfy the logarithmic Sobolev inequality with a given constant κ > 0. We discuss \(\mathcal{L}S_{lc}(\kappa )\) from a geometric point of view and we focus on related open questions.

On the Distribution of the ψ2-Norm of Linear Functionals on Isotropic Convex Bodies

It is known that every isotropic convex body K in \({\mathbb{R}}^{n}\) has a “subgaussian” direction with constant \(r\,=\,O(\sqrt{\log n})\). This follows from the upper bound \(\vert {\Psi }_{2}(K){\vert }^{1/n}\,\leq \,\frac{c\sqrt{\log n}} {\sqrt{n}} {L}_{K}\) for the volume of the body Ψ 2(K) with support function \({h}_{{\Psi }_{2}(K)}(\theta ) :{=\sup }_{2\leq q\leq n}\frac{\|\langle \cdot,{\theta \rangle \|}_{q}} {\sqrt{q}}\). The approach in all the related works does not provide estimates on the measure of directions satisfying a ψ2-estimate with a given constant r. We introduce the function \({\psi }_{K}(t) := \sigma (\{\theta \in {S}^{n-1} : {h}_{{\Psi }_{2}(K)}(\theta )\leq \mathit{ct}\sqrt{\log n}{L}_{K}\})\) and we discuss lower bounds for ψ K (t), \(t\geq 1\). Information on the distribution of the ψ2-norm of linear functionals is closely related to the problem of bounding from above the mean width of isotropic convex bodies.

A Gaussian small deviation inequality for convex functions

Let

_{}-estimates for marginals of log-concave probability measures

Z

Inequalities for the surface area of projections of convex bodies

be an

On the tightness of Gaussian concentration for convex functions

n

Neighborhoods on the Grassmannian of marginals with bounded isotropic constant

-dimensional Gaussian vector and let

Variance estimates and almost Euclidean structure

f: \mathbb R^n \to \mathbb R