  We prove an isoperimetric inequality for the uniform measure on a uniformly
convex body and for a class of uniformly log-concave measures (that we
introduce). These inequalities imply (up to universal constants) the
log-Sobolev inequalities proved by Bobkov--Ledoux as well as the isoperimetric
inequalities due to Bakry-Ledoux and Bobkov--Zegarlinski. We also recover a
concentration inequality for uniformly convex bodies, similar to that proved by
Gromov--Milman.
