Log-concave measures
Denis Feyel, A. Suleyman Ustunel
TL;DR
This paper explores log-concave measures, their properties, and subclasses like super log-concave measures, highlighting their stability, connections to measure transportation, and applications in inequalities and equations.
Contribution
It introduces super log-concave measures, analyzes their stability, and links log-concavity to measure transportation and Monge-Ampère equations.
Findings
Super log-concave measures satisfy a logarithmic Sobolev inequality.
Stability results for classes of log-concave measures are established.
Connections between log-concavity, measure transportation, and Monge-Ampère equations are demonstrated.
Abstract
We study the log-concave measures, their characterization via the Pr\'ekopa-Leindler property and also define a subset of it whose elements are called super log-concave measures which have the property of satisfying a logarithmic Sobolev inequality. We give some results about their stability. Certain relations with measure transportation of Monge-Kantorovitch and the Monge-Amp\'ere equation are also indicated with applications.
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Taxonomy
TopicsGeometry and complex manifolds · Geometric Analysis and Curvature Flows · Nonlinear Partial Differential Equations