Prekopa-Leindler Inequality Research Articles

Transport proofs of some discrete variants of the Prékopa-Leindler inequality

We give a transport proof of a discrete version of the displacement convexity of entropy on integers (Z), and get, as a consequence, two discrete forms of the Prekopa-Leindler Inequality : the Four Functions Theorem of Ahlswede and Daykin on the discrete hypercube [1] and a recent result on Z due to Klartag and Lehec [16].

Some extensions of the Prékopa–Leindler inequality using Borell’s stochastic approach

We present an abstract form of the Prekopa–Leindler inequality that includes several known —and a few new— related functional inequalities on Euclidean spaces. The method of proof and also the formulation of the new inequalities are based on Christer Borell’s stochastic approach to Brunn–Minkowski type inequalities.

Heat Equation and Convolution Inequalities

It is known that many classical inequalities linked to convolutions can be obtained by looking at the monotonicity in time of convolutions of powers of solutions to the heat equation, provided that both the exponents and the coefficients of diffusions are suitably chosen and related. This idea can be applied to give an alternative proof of the sharp form of the classical Young’s inequality and its converse, to Brascamp–Lieb type inequalities, Babenko’s inequality and Prekopa–Leindler inequality as well as the Shannon’s entropy power inequality. This note aims in presenting new proofs of these results, in the spirit of the original arguments introduced by Stam [35] to prove the entropy power inequality.

The Steiner symmetrization of log-concave functions and its applications

In this paper, we give a new definition of functional Steiner symmetrizations on logconcave functions. Using the functional Steiner symmetrization, we give a new proof of the classical Prekopa-Leindler inequality on log-concave functions. Mathematics subject classification (2010): 46E30, 52A40.

STABILITY OF THE PRÉKOPA–LEINDLER INEQUALITY

We prove a stability version of the Pr\'ekopa-Leindler inequality.

Stability of some versions of the Prékopa–Leindler inequality

Two consequences of the stability version of the one dimensional Prékopa–Leindler inequality are presented. One is the stability version of the Blaschke–Santaló inequality, and the other is a stability version of the Prékopa– Leindler inequality for even functions in higher dimensions, where a recent stability version of the Brunn–Minkowski inequality is also used in an essential way.

Partitions and functional Santaló inequalities

We give a direct proof of a functional Santalo inequality due to Fradelizi and Meyer. This provides a new proof of the Blaschke-Santalo inequality. The argument combines a logarithmic form of the Prekopa-Leindler inequality and a partition theorem of Yao and Yao.

From the Prékopa-Leindler inequality to modified logarithmic Sobolev inequality

We develop in this paper an improvement of the method given by S. Bobkov and M. Ledoux. Using the Prekopa-Leindler inequality, we prove a modified logarithmic Sobolev inequality adapted for all measures on $\dR^n$, with a strictly convex and super-linear potential. This inequality implies modified logarithmic Sobolev inequality for all uniform strictly convex potential as well as the Euclidean logarithmic Sobolev inequality.