An isoperimetric inequality for uniformly log-concave measures and uniformly convex bodies

We prove an isoperimetric inequality for uniformly log-concave measures and for the uniform measure on a uniformly convex body. These inequalities imply the log-Sobolev inequalities proved by Bobkov and Ledoux [12] and Bobkov and Zegarlinski [13]. We also recover a concentration inequality for uniformly convex bodies, similar to that proved by Gromov and Milman [22].