# Sharp isoperimetric inequalities and model spaces for the Curvature-Dimension-Diameter condition

### Emanuel Milman

Technion - Israel Institute of Technology, Haifa, Israel

## Abstract

We obtain new sharp isoperimetric inequalities on a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and generalized dimension and diameter of the convex support are bounded from above (possibly infinitely). Our inequalities are *sharp* for sets of any given measure and with respect to all parameters (curvature, dimension and diameter). Moreover, for each choice of parameters, we identify the *model spaces* which are extremal for the isoperimetric problem. In particular, we recover the Gromov–Lévy and Bakry–Ledoux isoperimetric inequalities, which state that whenever the curvature is strictly *positively* bounded from below, these model spaces are the $n$-sphere and Gauss space, corresponding to generalized dimension being $n$ and $∞$, respectively. In all other cases, which seem new even for the classical Riemannian-volume measure, it turns out that there is no single model space to compare to, and that a simultaneous comparison to a natural *one-parameter family* of model spaces is required, nevertheless yielding a sharp result.

## Cite this article

Emanuel Milman, Sharp isoperimetric inequalities and model spaces for the Curvature-Dimension-Diameter condition. J. Eur. Math. Soc. 17 (2015), no. 5, pp. 1041–1078

DOI 10.4171/JEMS/526