Dario Cordero-Erausquin

Invariances in variance estimates

We provide variants and improvements of the Brascamp‐Lieb variance inequality which takes into account the invariance properties of the underlying measure. This is applied to spectral gap estimates for log-concave measures with many symmetries and to non-interacting conservative spin systems.

Santaló's inequality on Cn by complex interpolation

A new approach to Santalos inequality on Cn is obtained by combining complex interpolation and Berndtssons generalization of Prekopas inequality. To cite this article: D. Cordero-Erausquin, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 767–772.

The geometry of Euclidean convolution inequalities and entropy

The goal of this paper is to show that some convolution type inequalities from Harmonic Analysis and Information Theory, such as Youngs convolution inequality (with sharp constant), Nelsons hypercontractivity of the Hermite semi-group or Shannons inequality, can be reduced to a simple geometric study of frames of ℝ 2 . We shall derive directly entropic inequalities, which were recently proved to be dual to the Brascamp-Lieb convolution type inequalities.

Hypercontractive Measures, Talagrand’s Inequality, and Influences

We survey several Talagrand type inequalities and their application to influences with the tool of hypercontractivity for both discrete and continuous, and product and non-product models. The approach covers similarly by a simple interpolation the framework of geometric influences recently developed by N. Keller, E. Mossel and A. Sen. Geometric Brascamp-Lieb decompositions are also considered in this context.

Interpolations, Convexity and Geometric Inequalities

We survey some interplays between spectral estimates of Hormander-type, degenerate Monge-Ampere equations and geometric inequalities related to log-concavity such as Brunn-Minkowski, Santalo or Busemann inequalities.

A transport inequality on the sphere obtained by mass transport

Using McCanns transportation map, we establish a transport inequality on compact manifolds with positive Ricci curvature. This inequality contains the sharp spectral comparison estimates.