Ricci curvature and functional inequalities on graphs

2012 ◽  
pp. 663-675
Keyword(s):  
Ricci Curvature ◽  
Functional Inequalities

scholarly journals Characterization of Pinched Ricci Curvature by Functional Inequalities

2017 ◽  
Vol 28 (3) ◽  
pp. 2312-2345 ◽  
Author(s):  
Li-Juan Cheng ◽  
Anton Thalmaier
Keyword(s):  
Ricci Curvature ◽  
Functional Inequalities

scholarly journals Dilation Type Inequalities for Strongly-Convex Sets in Weighted Riemannian Manifolds

2021 ◽  
Vol 9 (1) ◽  
pp. 219-253
Author(s):  
Hiroshi Tsuji
Keyword(s):  
Ricci Curvature ◽  
Convex Sets ◽  
Convex Geometry ◽  
Type Inequality ◽  
Model Space ◽  
Functional Inequalities ◽  
Curvature Bounds ◽  
Strongly Convex ◽  
Weighted Ricci Curvature ◽  
The One

Abstract In this paper, we consider a dilation type inequality on a weighted Riemannian manifold, which is classically known as Borell’s lemma in high-dimensional convex geometry. We investigate the dilation type inequality as an isoperimetric type inequality by introducing the dilation profile and estimate it by the one for the corresponding model space under lower weighted Ricci curvature bounds. We also explore functional inequalities derived from the comparison of the dilation profiles under the nonnegative weighted Ricci curvature. In particular, we show several functional inequalities related to various entropies.

ON THE COMPACTNESS OF MANIFOLDS

2003 ◽  
Vol 06 (supp01) ◽  
pp. 29-38 ◽  
Author(s):  
XUE-MEI LI ◽  
FENG-YU WANG
Keyword(s):  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Positive Constant ◽  
Riemannian Metric ◽  
Sobolev Inequalities ◽  
Functional Inequalities ◽  
The Family ◽  
Bounded Below

It is believed that the family of Riemannian manifolds with negative curvatures is much richer than that with positive curvatures. In fact there are many results on the obstruction of furnishing a manifold with a Riemannian metric whose curvature is positive. In particular any manifold admitting a Riemannian metric whose Ricci curvature is bounded below by a positive constant must be compact. Here we investigate such obstructions in terms of certain functional inequalities which can be considered as generalized Poincaré or log-Sobolev inequalities. A result of Saloff-Coste is extended.

scholarly journals Functional Inequalities for Two-Level Concentration

2021 ◽  
Author(s):  
Franck Barthe ◽  
Michał Strzelecki
Keyword(s):  
Sobolev Inequality ◽  
Poincaré Inequality ◽  
Logarithmic Sobolev Inequality ◽  
Probability Measures ◽  
Exponential Rate ◽  
Functional Inequalities ◽  
Concentration Inequality ◽  
Poincare Inequality ◽  
Free Concentration ◽  
Concentration Property

AbstractProbability measures satisfying a Poincaré inequality are known to enjoy a dimension-free concentration inequality with exponential rate. A celebrated result of Bobkov and Ledoux shows that a Poincaré inequality automatically implies a modified logarithmic Sobolev inequality. As a consequence the Poincaré inequality ensures a stronger dimension-free concentration property, known as two-level concentration. We show that a similar phenomenon occurs for the Latała–Oleszkiewicz inequalities, which were devised to uncover dimension-free concentration with rate between exponential and Gaussian. Motivated by the search for counterexamples to related questions, we also develop analytic techniques to study functional inequalities for probability measures on the line with wild potentials.

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