Matthieu Fradelizi

Bio: Matthieu Fradelizi is an academic researcher from University of Paris. The author has contributed to research in topics: Convex body & Convex hull. The author has an hindex of 18, co-authored 57 publications receiving 1101 citations. Previous affiliations of Matthieu Fradelizi include Paris 12 Val de Marne University & Centre national de la recherche scientifique.

Some functional forms of Blaschke–Santaló inequality

Abstract: We establish new functional versions of the Blaschke?Santalo inequality on the volume product of a convex body which generalize to the non-symmetric setting an inequality of Ball [Isometric problems in l p and sections of convex sets. PhD Dissertation, Cambridge, 1986] and we give a simple proof of the case of equality. As a corollary, we get some inequalities for log-concave functions and Legendre transforms which extend the recent result of Artstein et al. [Mathematika 51:33?48, 2004], with its equality case

Sections of convex bodies through their centroid

Abstract: We give the best estimate in the comparison of the volume of the section of a convex body in \( {\Bbb R}^n \) through its centroid by a k-dimensional affine subspace Ek with the volume of the section by any affine subspace parallel to Ek.

Some functional forms of Blaschke-Santal\'o inequality

Abstract: We establish new functional versions of the Blaschke-Santal\'o inequality on the volume product of a convex body which generalize to the non-symmetric setting an inequality of K. Ball and we give a simple proof of the case of equality. As a corollary, we get some inequalities for $\log$-concave functions and Legendre transforms which extend the recent result of Artstein, Klartag and Milman, with its equality case.

The Extreme Points of Subsets of s -Concave Probabilities and a Geometric Localization Theorem

Abstract: We prove that the extreme points of the set of s-concave probability measures satisfying a linear constraint are some Dirac measures and some s-affine probabilities supported by a segment. From this we deduce that the constrained maximization of a convex functional on the s-concave probability measures is reduced to this small set of extreme points. This gives a new approach to a localization theorem due to Kannan, Lovasz and Simonovits which happens to be very useful in geometry to obtain inequalities for integrals like concentration and isoperimetric inequalities. Roughly speaking, the study of such inequalities is reduced to these extreme points.

Isoperimetric and Analytic Inequalities for Log-Concave Probability Measures

Abstract: We discuss an approach, based on the Brunn–Minkowski inequality, to isoperimetric and analytic inequalities for probability measures on Euclidean space with logarithmically concave densities. In particular, we show that such measures have positive isoperimetric constants in the sense of Cheeger and thus always share Poincare-type inequalities. We then describe those log-concave measures which satisfy isoperimetric inequalities of Gaussian type. The results are precised in dimension 1.

On convex perturbations with a bounded isotropic constant

Abstract: Let $$ K \subset {\user2{\mathbb{R}}}^{n} $$ be a convex body and ɛ > 0. We prove the existence of another convex body $$ K' \subset {\user2{\mathbb{R}}}^{n} $$ , whose Banach–Mazur distance from K is bounded by 1 + ɛ, such that the isotropic constant of K’ is smaller than $$ c \mathord{\left/ {\vphantom {c {{\sqrt \varepsilon }}}} \right. \kern- ulldelimiterspace} {{\sqrt \varepsilon }} $$ , where c > 0 is a universal constant. As an application of our result, we present a slight improvement on the best general upper bound for the isotropic constant, due to Bourgain.