Matthieu Fradelizi

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, Lovász 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.

Some inequalities about mixed volumes

AbstractWe prove inequalities about the quermassintegralsVk(K) of a convex bodyK in ℝn (here,Vk(K) is the mixed volumeV((K, k), (Bn,n − k)) whereBn is the Euclidean unit ball). (i) The inequality

A Short Solution to the Busemann-Petty Problem

\frac{{V_k \left( {K + L} \right)}}{{V_{k - 1} \left( {K + L} \right)}} \geqslant \frac{{V_k \left( K \right)}}{{V_{k - 1} \left( K \right)}} + \frac{{V_k \left( L \right)}}{{V_{k - 1} \left( L \right)}}

On the analogue of the concavity of entropy power in the Brunn–Minkowski theory ☆

holds for every pair of convex bodiesK andL in ℝn if and only ifk=2 ork=1. (ii) Let 0≤k≤p≤n. Then, for everyp-dimensional subspaceE of ℝn,

An Application of Shadow Systems to Mahler’s Conjecture

\frac{{V_{n - k} \left( K \right)}}{{\left| K \right|}} \geqslant \frac{1}{{\left( {_{n - p}^{n - p + k} } \right)}}\frac{{V_{p - k} \left( {P_E K} \right)}}{{\left| {P_E K} \right|}},

Information concentration for convex measures

wherePEK denotes the orthogonal projection ofK ontoE. The proof is based on a sharp upper estimate for the volume ratio |K|/|L| in terms ofVn−k(K)/Vn−k(L), wheneverL andK are two convex bodies in ℝn such thatK ⊆L.