Isoperimetric and analytic inequalities for log-concave probability measures

We discuss an approach, based on the Brunn-Minkowski inequality, to isoperi metric and analytic inequalities for probability measures on Euclidean spac e with logarithmically concave densities. In particular, we show that such measures have positive isoperimetric constants in the sense of Cheeger and thus always share Poincare-type inequalities. We then describe those log-co ncave measures which satisfy isoperimetric inequalities of Gaussian type. T he results are precised in dimension 1.