Sharp Isoperimetric Inequalities and Model Spaces for Curvature-Dimension-Diameter Condition

By Emanuel Milman. 2011

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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--L\'evy 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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b49500f6345c81d6476bcc346f909af2281fcda2

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Characterization of Low Dimensional RCD*(K, N) Spaces (2015, background): “(5.5) where, IK,N,D(·) is the Milman’s model isoperimetric profile (see Cavalletti-Mondino [9] and Milman [31] for the precise definitions).”
Brascamp–Lieb-Type Inequalities on Weighted Riemannian Manifolds with Boundary (2013, methodology): “In practice, when N ≥ n (and thus β < ∞), we need to ensure that the resulting compact weighted manifold is smooth at its vertices (at t ∈ {−β, β} in Case (1) and t = β in Case (2)), and this is achieved as in [33] by gluing appropriate caps.”
Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds (2015, methodology): “Milman [48] who extended the Lévy-Gromov isoperimetric inequality to smooth manifolds with densities, i.”
The globalization theorem for the Curvature-Dimension condition (2016, background): “The localization paradigm, developed by Payne–Weinberger [62], Gromov–Milman [44] and Kannan–Lovász–Simonovits [46], is a powerful tool to reduce various analytic and geometric inequalities on the space (Rn, d,m) to appropriate one-dimensional counterparts.”
Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature (2020, methodology): “An extension of Milman’s results is due to Cavalletti and Mondino [13], who used the OMT-approach in combination to themeasure disintegrationmethod of Klartag [33] to prove a sharp Lévy-Gromov-type isoperimetric inequality on metric measure spaces satisfying the CD(κ, N ) condition with κ > 0 and N ”
Poincaré and Brunn-Minkowski inequalities on the boundary of weighted Riemannian manifolds (2017, background): “also [64, 38, 20, 21, 77, 6, 78, 57, 51, 5] and the references therein), especially since Perelman’s work on the Poincaré Conjecture [62], and the extension of the Curvature-Dimension condition to the metric-measure space setting by Lott–Sturm–Villani [72, 46].”