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.