JournalsjemsVol. 17 , No. 5pp. 1041–1078

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

  • Emanuel Milman

    Technion - Israel Institute of Technology, Haifa, Israel
Sharp isoperimetric inequalities and model spaces for the Curvature-Dimension-Diameter condition cover
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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 nn-sphere and Gauss space, corresponding to generalized dimension being nn and \infty, 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