Andreas Malliaris (Univ. Lille) : Functional and isoperimetric inequalities in product spaces

Let $(X,d,\mu)$ be a metric probability metric space. We are interested in isoperimetric inequalities for the product space $(X^n,\mu^n)$ equipped with the $\ell_q$ product metric, $q\geq 2$. By employing the functional approach of Bobkov for dimension free Gaussian isoperimetry we recover a result of E. Milman for product probability spaces equipped with the $\ell_2$ metric and extend it to $\ell_q$ metrics. The crux of this analytic approach is to understand how to deal with more complicated energy terms compared to classical Sobolev type inequalities. This is joint work with Franck Barthe.