A Mass Transportation Approach to Quantitative Isoperimetric Inequalities

Some Methods for Studying Stability in Isoperimetric Type Problems

We review the method of quantitative symmetrization inequalities introduced in Fusco, Maggi and Pratelli, “The sharp quantitative isoperimetric inequality”, Ann. of Math. Introduction Our aim is to introduce certain ideas related to the quantitative study of isoperimetric type inequalities. The main efforts are devoted to the proof of a sharp quantitative version of the Euclidean isoperimetric ...

Log-sobolev, Isoperimetry and Transport Inequalities on Graphs

In this paper, we study some functional inequalities (such as Poincaré inequalities, logarithmic Sobolev inequalities, generalized Cheeger isoperimetric inequalities, transportation-information inequalities and transportation-entropy inequalities) for reversible nearest-neighbor Markov processes on a connected finite graph by means of (random) path method. We provide estimates of the involved c...

Transportation-information Inequalities for Markov Processes (ii) : Relations with Other Functional Inequalities

We continue our investigation on the transportation-information inequalities WpI for a symmetric markov process, introduced and studied in [14]. We prove that WpI implies the usual transportation inequalities WpH, then the corresponding concentration inequalities for the invariant measure μ. We give also a direct proof that the spectral gap in the space of Lipschitz functions for a diffusion pr...

Quantitative Isoperimetric Inequalities in H

In the Heisenberg group H, n ≥ 1, we prove quantitative isoperimetric inequalities for Pansu’s spheres, that are known to be isoperimetric under various assumptions. The inequalities are shown for suitably restricted classes of competing sets and the proof relies on the construction of sub-calibrations.

Some Higher Order Isoperimetric Inequalities via the Method of Optimal Transport