We discuss an approach, based on the Brunn–Minkowski inequality, to isoperimetric and analytic inequalities for probability measures on Euclidean space with logarithmically concave densities. In particular, we show that such measures have positive isoperimetric constants in the sense of Cheeger and thus always share Poincaré-type inequalities. We then describe those log-concave measures which satisfy isoperimetric inequalities of Gaussian type. The results are precised in dimension 1.
"Isoperimetric and Analytic Inequalities for Log-Concave Probability Measures." Ann. Probab. 27 (4) 1903 - 1921, October 1999. https://doi.org/10.1214/aop/1022874820