arXiv math/05034762005-03-23math.FAmath.PR

Logarithmic Sobolev inequality for log-concave measure from Prekopa-Leindler inequality

Ivan Gentil

英文摘要：

We develop in this paper an amelioration of the method given by S. Bobkov and M. Ledoux in GAFA (2000). We prove by Prekopa-Leindler Theorem an optimal modified logarithmic Sobolev inequality adapted for all log-concave measure on $\dR^n$. This inequality implies results proved by Bobkov and Ledoux, the Euclidean Logarithmic Sobolev inequality generalized in the last years and it also implies some convex logarithmic Sobolev inequalities for large entropy.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Some extensions of the Prékopa-Leindler inequality using Borell's stochastic approach

We present an abstract form of the Prékopa-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.

2016-10-24math.FAmath.PR

Rearrangement and Prekopa-Leindler type inequalities

We investigate the interactions of functional rearrangements with Prekopa-Leindler type inequalities. It is shown that that a general class of integral inequalities tighten on rearrangement to "isoperimetric" sets with respect to a relevant measure. Applications to the Borell-Brascamp-Lieb, Borell-Ehrhart, and the recent polar Prekopa-Leindler inequalities are demonstrated. It is also proven that an integrated form of the Gaussian log-Sobolev inequality decreases on half-space rearrangement.

2019-05-22cs.ITmath.FAmath.IT

A Prékopa-Leindler type inequality of the $L_p$ Brunn-Minkowski inequality

In this paper, we prove a Prékopa-Leindler type inequality of the $L_p$ Brunn-Minkowski inequality. It extends an inequality proved by Das Gupta [8] and Klartag [16], and thus recovers the Prékopa-Leindler inequality. In addition, we prove a functional $L_p$ Minkowski inequality.

2021-03-24math.FAmath.MG

Borell's generalized Prékopa-Leindler inequality: A simple proof

We present a simple proof of Christer Borell's general inequality in the Brunn-Minkowski theory. We then discuss applications of Borell's inequality to the log-Brunn-Minkowski inequality of Böröczky, Lutwak, Yang and Zhang.

2015-12-14math.FA

On a linear refinement of the Prékopa-Leindler inequality

If $f,g:\mathbb{R}^n\longrightarrow\mathbb{R}_{\geq0}$ are non-negative measurable functions, then the Prékopa-Leindler inequality asserts that the integral of the Asplund sum (provided that it is measurable) is greater or equal than the $0$-mean of the integrals of $f$ and $g$. In this paper we prove that under the sole assumption that $f$ and $g$ have a common projection onto a hyperplane, the Prékopa-Leindler inequality admits a linear refinement. Moreover, the same inequality can be obtained when assuming that both projections (not necessarily equal as functions) have the same integral. An…

2015-03-28math.FA

A quantitative stability result for the Prékopa-Leindler inequality for arbitrary measurable functions

We prove that if a triplet of functions satisfies almost equality in the Prékopa-Leindler inequality, then these functions are close to a common log-concave function, up to multiplication and rescaling. Our result holds for general measurable functions in all dimensions, and provides a quantitative stability estimate with computable constants.

2022-01-27math.FAmath.MG

更多相关论文收起更多论文

Prékopa-Leindler 不等式的改进稳定性版本

Improved stability versions of the Prékopa-Leindler inequality

本文利用径向递减函数间传输映射与泛函迹不等式，得到 Prékopa-Leindler 不等式在维数和对数凹性参数上均一致的稳定性指数，并证明一维及高维径向情形的尖锐稳定性。

2024-10-01math.FAmath.MG

On a geometric combination of functions related to Prékopa-Leindler inequality

We introduce a new operation between nonnegative integrable functions on $\mathbb{R} ^n$, that we call geometric combination; it is obtained via a mass transportation approach, playing with inverse distribution functions. The main feature of this operation is that the Lebesgue integral of the geometric combination equals the geometric mean of the two separate integrals; as a natural consequence, we derive a new functional inequality of Prékopa-Leindler type. When applied to the characteristic functions of two measurable sets, their geometric combination provides a set whose volume equals the g…

2022-04-25math.FA

Prékopa-Leindler 不等式与 Borell-Brascamp-Lieb 不等式的尖锐定量稳定性

Sharp Quantitative Stability for the Prékopa-Leindler and Borell-Brascamp-Lieb Inequalities

ETH Zürich（苏黎世联邦理工学院）；Ecole Polytechnique Fédérale Lausanne（洛桑联邦理工学院）

本文构建统一框架，在完全一般条件下证明 Borell-Brascamp-Lieb 不等式的最优定量稳定性，并据此解决 Prékopa-Leindler 不等式的尖锐稳定性猜想。

2026-10-08math.APmath.COmath.FA

Prékopa-Leindler不等式的新无穷小形式及其乘法结构与应用

A new infinitesimal form of the Prékopa-Leindler inequality with multiplicative structure and applications

本文提出一种新的Prékopa-Leindler不等式形式，利用其乘法结构开发替代L_2方法，推导稳定性估计并研究Brunn-Minkowski猜想。

2026-02-26math.FAmath.MG

卷积下对数超模ularity的稳定性

The stability of log-supermodularity under convolution

本文研究了对数超模ularity函数在卷积下的稳定性，并验证了相关猜想，同时提供了两个定理的传输证明及新的泛化结果。

2025-12-22cs.ITmath.FAmath.IT

Poisson processes and a log-concave Bernstein theorem

We discuss interplays between log-concave functions and log-concave sequences. We prove a Bernstein-type theorem, which characterizes the Laplace transform of log-concave measures on the half-line in terms of log-concavity of the alternating Taylor coefficients. We establish concavity inequalities for sequences inspired by the Prékopa-Leindler and the Walkup theorems. One of our main tools is a stochastic variational formula for the Poisson average.

2018-07-09math.FAmath.PR

Log-concave measures

We study the log-concave measures, their characterization via the Prékopa-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-Ampére equation are also indicated with applications.

2010-05-27math.FAmath.PR

Displacement convexity of entropy and related inequalities on graphs

We introduce the notion of an interpolating path on the set of probability measures on finite graphs. Using this notion, we first prove a displacement convexity property of entropy along such a path and derive Prekopa-Leindler type inequalities, a Talagrand transport-entropy inequality, certain HWI type as well as log-Sobolev type inequalities in discrete settings. To illustrate through examples, we apply our results to the complete graph and to the hypercube for which our results are optimal -- by passing to the limit, we recover the classical log-Sobolev inequality for the standard Gaussian …

2012-07-21math.FAmath.PR

On matrix-valued log-concavity and related Prekopa and Brascamp-Lieb inequalities

We propose a new, self-contained, approach to H. Raufi's extension of Prekopa's theorem for matrix-valued log-concave functions. Along the way, new related inequalities are established, in particular a Brascamp-Lieb variance inequality for matrix weights.

2018-01-15math.APmath.FA

Blaschke-Santalo inequality for many functions and geodesic barycenters of measures

Motivated by the geodesic barycenter problem from optimal transportation theory, we prove a natural generalization of the Blaschke-Santalo inequality and the affine isoperimetric inequalities for many sets and many functions. We derive from it an entropy bound for the total Kantorovich cost appearing in the barycenter problem. We also establish a "pointwise Prekopa-Leindler inequality" and show a monotonicity property of the multimarginal Blaschke-Santalo functional.

2021-07-24math.FA

加权边缘分布的凹性原理

Concavity principles for weighted marginals

本文通过局部方法建立加权边缘凹性的一般框架，获得旋转不变对数凹测度的维数Brunn-Minkowski不等式函数版本及包含B-不等式的Prékopa型凹性原理。

2025-06-20math.FAmath.MG

Geometry of log-concave functions: the $L_p$ Asplund sum and the $L_{p}$ Minkowski problem

The aim of this paper is to develop a basic framework of the $L_p$ theory for the geometry of log-concave functions, which can be viewed as a functional "lifting" of the $L_p$ Brunn-Minkowski theory for convex bodies. To fulfill this goal, by combining the $L_p$ Asplund sum of log-concave functions for all $p>1$ and the total mass, we obtain a Prékopa-Leindler type inequality and propose a definition for the first variation of the total mass in the $L_p$ setting. Based on these, we further establish an $L_p$ Minkowski type inequality related to the first variation of the total mass and derive …

2020-06-30math.APmath.FAmath.MG

Brunn-Minkowski inequalities in product metric measure spaces

Given one metric measure space $X$ satisfying a linear Brunn-Minkowski inequality, and a second one $Y$ satisfying a Brunn-Minkowski inequality with exponent $p\ge -1$, we prove that the product $X\times Y$ with the standard product distance and measure satisfies a Brunn-Minkowski inequality of order $1/(1+p^{-1})$ under mild conditions on the measures and the assumption that the distances are strictly intrinsic. The same result holds when we consider restricted classes of sets. We also prove that a linear Brunn-Minkowski inequality is obtained in $X\times Y$ when $Y$ satisfies a Prékopa-Leind…

2017-05-04math.FAmath.MG

On $L_p$-Brunn-Minkowski type and $L_p$-isoperimetric type inequalities for general measures

In 2011 Lutwak, Yang and Zhang extended the definition of the $L_p$-Minkowski convex combination ($p \geq 1$) introduced by Firey in the 1960s from convex bodies containing the origin in their interiors to all measurable subsets in $\mathbb{R}^n$, and as a consequence, extended the $L_p$-Brunn-Minkowski inequality ($L_p$-BMI) to the setting of all measurable sets. In this paper, we present a functional extension of their $L_p$-Minkowski convex combination---the $L_{p,s}$--supremal convolution and prove the $L_p$-Borell-Brascamp-Lieb type ($L_p$-BBL) inequalities. Based on the $L_p$-BBL type in…

2020-06-08math.FA

径向p次均体是凸的

Convexity of Radial Mean Bodies via an Extension of Ball's Bodies

Carnegie Mellon University（卡内基梅隆大学）

本文利用Prékopa定理证明了当p∈(-1,0)时径向p次均体的凸性，同时给出了Keith Ball定理的新证明并扩展至p负区间。

2026-07-07math.FAmath.MG