Rearrangement and Prékopa–Leindler Type Inequalities

Abstract

We investigate the interactions of functional rearrangements with Prékopa-Leindler type inequalities. It is shown that that certain set theoretic rearrangement inequalities can be lifted to functional analogs, thus demonstrating that several important integral inequalities tighten on functional rearrangement about "isoperimetric" sets with respect to a relevant measure. Applications to the Borell-Brascamp-Lieb, Borell-Ehrhard, and the recent polar Prékopa-Leindler inequalities are demonstrated. It is also proven that an integrated form of the Gaussian log-Sobolev inequality sharpens on rearrangement.

On the Entropy Power Inequality for the Rényi Entropy of Order [0, 1]

“…Motivated by this fact the authors replaced the exponential distribution in the example above with its spherically symmetric rearrangement, the Laplace distribution, to yield a tighter lower bound in an announcement of this work [34]. Additionally, since spherically symmetric rearrangement is stable on the class of log-concave random vectors (see [35,Corollary 5.2]), one can reduce to random vectors with spherically symmetric decreasing densities, even under the log-concave restriction taken in this work.…”

A discrete complement of Lyapunov's inequality and its information theoretic consequences

“…1 By Lyapunov's inequality we refer to the fact that p → log f p p is convex in p for a general measurable function f and measure. 2 Note that for continuous variables on R, proving the result for monotone variables is equivalent to the general result since log-concavity is preserved under rearrangement, see for example [31].…”

A stochastic Prekopa-Leindler inequality for log-concave functions

“…Brascamp and Lieb proved (1.6) via rearrangement inequalities through the reverse Young inequality [9]. More recently, Melbourne [35] derived families of rearrangement inequalities refining (1.6); the version we study here involves Borel measurable f, g : R n → [0, ∞), 0 < λ < 1 and the sup-convolution…”