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

Journal of the European Mathematical Society (2015)

- Volume: 017, Issue: 5, page 1041-1078
- ISSN: 1435-9855

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topMilman, Emanuel. "Sharp isoperimetric inequalities and model spaces for the Curvature-Dimension-Diameter condition." Journal of the European Mathematical Society 017.5 (2015): 1041-1078. <http://eudml.org/doc/277164>.

@article{Milman2015,

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 $\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.},

author = {Milman, Emanuel},

journal = {Journal of the European Mathematical Society},

keywords = {isoperimetric inequality; generalized Ricci tensor; manifold with density; geodesically convex; model space; isoperimetric inequality; generalized Ricci tensor; manifold with density; geodesically convex; model space},

language = {eng},

number = {5},

pages = {1041-1078},

publisher = {European Mathematical Society Publishing House},

title = {Sharp isoperimetric inequalities and model spaces for the Curvature-Dimension-Diameter condition},

url = {http://eudml.org/doc/277164},

volume = {017},

year = {2015},

}

TY - JOUR

AU - Milman, Emanuel

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

JO - Journal of the European Mathematical Society

PY - 2015

PB - European Mathematical Society Publishing House

VL - 017

IS - 5

SP - 1041

EP - 1078

AB - 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 $\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.

LA - eng

KW - isoperimetric inequality; generalized Ricci tensor; manifold with density; geodesically convex; model space; isoperimetric inequality; generalized Ricci tensor; manifold with density; geodesically convex; model space

UR - http://eudml.org/doc/277164

ER -

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