Aldo Pratelli

The sharp Sobolev inequality in quantitative form

A quantitative version of the sharp Sobolev inequality in W (R), 1 < p < n, is established with a remainder term involving the distance from extremals.

On the isoperimetric deficit in Gauss space

<abstract abstract-type=TeX><p>We prove a sharp quantitative version of the isoperimetric inequality in the space

Integral estimates for transport densities

{Bbb R}^n

A note on Cheeger sets

endowed with the Gaussian measure.

On a conjecture by Auerbach

We introduce some integration-by-parts methods that improve upon the L p estimates on transport densitites from the recent paper by De Pascale–Pratelli [DP-P].

A geometric approach to correlation inequalities in the plane

Starting from the quantitative isoperimetric inequality, we prove a sharp quantitative version of the Cheeger inequality.

On the boundary of the attainable set of the dirichlet spectrum

In 1938 Herman Auerbach published a paper where he showed a deep connection between the solutions of the Ulam problem of floating bodies and a class of sets studied by Zindler, that are the planar sets whose bisecting chords have all the same length. In the same paper he conjectured that among Zindler sets the one with minimal area, as well as with maximal perimeter, is given by the so-called “Auerbach triangle”. We prove here that his conjecture was true.

Finite element approximation of the Sobolev constant

By elementary geometric arguments, correlation inequalities for radially symmetric probability measures are proved in the plane. Precisely, it is shown that the correlation ratio for pairs of width-decreasing sets is minimized within the class of infinite strips. Since open convex sets which are symmetric with respect to the origin turn out to be width-decreasing sets, Pitt’s Gaussian correlation inequality (the two-dimensional case of the long-standing Gaussian correlation conjecture) is derived as a corollary, and it is in fact extended to a wide class of radially symmetric measures. Résumé. En utilisant des arguments géométriques élémentaires, on démontre des inégalités de corrélation pour des mesures de probabilité à symétrie radiale. Plus précisément on montre que, parmi la famille des ensembles width-decreasing, le ratio de corrélation est minimisé par des bandes. Comme les ouverts convexes symétriques appartiennent à cette famille, on retrouve comme corollaire le résultat de Pitt sur la validité de la conjecture de corrélation gaussiennne en dimension 2, qui est étendue dans ce papier à une large classe de mesures à symétrie radiale.

A mass transportation approach to quantitative isoperimetric inequalities

Denoting by