Isoperimetric and Analytic Inequalities for Log-Concave Probability Measures

Abstract

We discuss an approach, based on the Brunn–Minkowski\ninequality, to isoperimetric and analytic inequalities for probability measures\non Euclidean space with logarithmically concave densities. In particular, we\nshow that such measures have positive isoperimetric constants in the sense of\nCheeger and thus always share Poincaré-type inequalities. We then\ndescribe those log-concave measures which satisfy isoperimetric inequalities of\nGaussian type. The results are precised in dimension 1.