arXiv subjects
Publications and source records attributed to Bernard Maurey.
We present an abstract form of the Prékopa-Leindler inequality that includes several known -and a few new- related functional inequalities on Euclidean spaces. The method of proof and also the formulation of the new inequalities are based on Christer Borell's stochastic approach to Brunn-Minkowski type inequalities.
This survey is based on a series of lectures given by the authors at the working seminar "Convexité et Probabilités" at UPMC Jussieu, Paris, during the spring 2013. It is devoted to maximal inequalities associated to symmetric convex sets in high dimensional linear spaces, a topic mainly developed between 1982 and 1990 but recently renewed by further advances. The series focused on proving for these maximal functions inequalities in $L^p(\mathbb{R}^n)$ with bounds independent of the dimension $n$, for all $p \in (1, +\infty]$ in the best cases. This program was initiated in 1982 by Elias Stein, who obtained the first theorem of this kind for the family of Euclidean balls in arbitrary dimension. We present several results along this line, proved by Bourgain, Carbery and Müller during the period 1986--1990, and a new one due to Bourgain (2014) for the family of cubes in arbitrary dimension. We complete the cube case with negative results for the weak type $(1, 1)$ constant, due to Aldaz, Aubrun and Iakovlev--Strömberg between 2009 and 2013.
Symmetric Monge-Kantorovich transport problems involving a cost function given by a family of vector fields were used by Ghoussoub-Moameni to establish polar decompositions of such vector fields into $m$-cyclically monotone maps composed with measure preserving $m$-involutions ($m\geq 2$). In this note, we relate these symmetric transport problems to the Brenier solutions of the Monge and Monge-Kantorovich problem, as well as to the Gangbo-Świȩch solutions of their multi-marginal counterparts, both of which involving quadratic cost functions.
A unified analytic solution to the Busemann-Petty problem was recently found by Gardner, Koldobsky and Schlumprecht. We give an elementary proof of their formulas for the inverse Radon transform.
In this paper structure of infinite dimensional Banach spaces is studied by using an asymptotic approach based on stabilization at infinity of finite dimensional subspaces which appear everywhere far away. This leads to notions of asymptotic structures and asymptotic versions of a given Banach space. As an example of application of this approach, a class of asymptotic $l_p$-spaces is introduced and investigated in detail. Some properties of this class, as duality and complementation, are analogous to properties of classical $l_p$ spaces, although the latter is more ``regular'' than its classical counterpart; in contrast, the property exhibited in the uniqueness theorem is very different than for spaces $l_p$.
In this note we show that every Banach space $X$ not containing $\ell_1^n$ uniformly and with unconditional basis contains an arbitrarily distortable subspace.
We construct a Banach space that does not contain any infinite unconditional basic sequence.
We introduce a concentration property for probability measures on $\scriptstyle{R^n}$, which we call Property~($\scriptstyleτ$); we show that this property has an interesting stability under products and contractions (Lemmas 1,~2,~3). Using property~($\scriptstyleτ$), we give a short proof for a recent deviation inequality due to Talagrand. In a third section, we also recover known concentration results for Gaussian measures using our approach.}