Isoperimetric and Concentration Inequalities - Equivalence under Curvature Lower Bound

By Emanuel Milman. 2009

82 citations recorded by the source corpus.

It is well known that isoperimetric inequalities imply in a very general measure-metric-space setting appropriate concentration inequalities. The former bound the boundary measure of sets as a function of their measure, whereas the latter bound the measure of sets separated from sets having half the total measure, as a function of their mutual distance. We show that under a lower bound condition on the Bakry--\'Emery curvature tensor of a Riemannian manifold equipped with a density, completely general concentration inequalities imply back their isoperimetric counterparts, up to dimension \emph{independent} bounds. As a corollary, we can recover and extend all previously known (dimension dependent) results by generalizing an isoperimetric inequality of Bobkov, and provide a new proof that under natural convexity assumptions, arbitrarily weak concentration implies a dimension independent linear isoperimetric inequality. Further applications will be described in a subsequent work. Contrary to previous attempts in this direction, our method is entirely geometric, continuing the approach set forth by Gromov and adapted to the manifold-with-density setting by Morgan.

View Paper: https://www.semanticscholar.org/paper/fd5371a857904f4b8eb73f449d7b823cb44044b2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Citations in the corpus

Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance (2017, background): “Establishing similar concentration results for other measures is closely connected to many topics in modern geometric probability, such as isoperimetry [36], logarithmic Sobolev inequalities [30], and so-called transportation inequalities, which link Wasserstein distances and the Kullback–Leibler di”
Sharp Isoperimetric Inequalities and Model Spaces for Curvature-Dimension-Diameter Condition (2011, background): “In [57, 58], we considered concentration inequalities, and showed that under the CD(ρ,∞) condition, these imply isoperimetric inequalities which are essentially best possible, up to dimension independent constants.”
Brascamp–Lieb-Type Inequalities on Weighted Riemannian Manifolds with Boundary (2013, background): “It is well-known that H g governs the ﬁrst variation of Vol ∂M under the normal-map t 7→ exp( tν ), and similarly H µ governs the ﬁrst variation of exp( V ) d Vol ∂M in the weighted-manifold setting, see e.g. [30].”
Transport Inequalities. A Survey (2010, unknown): “In a series of papers [85–87], E.”
Beyond traditional Curvature-Dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension (2014, background): “When N < 0, this is witnessed by the CD(0 , The case N ∈ [ n, ∞ ] has already been treated in the literature; see e.g. [43].”