The BrunnMinkowski theorem and related geometric and functional inequalities
The BrunnMinkowski theorem and related geometric and functional inequalities
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DOI：
10.4171/022-2/72
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发表时间：
2006
期刊：
影响因子：
--
通讯作者：
F. Barthe
中科院分区：
文献类型：
--
作者：
F. Barthe
Brunn - Minkowski不等式给出了和集的勒贝格测度的下界。它在凸体理论中起着至关重要的作用。这个主题与等渗法或功能分析有许多相互作用。我们在这里的目的是报告这些涉及最优质量传递或热方程的相互作用的一些最新方面。除其他内容外，我们将介绍平坦集或高斯空间中的布吕恩-闵可夫斯基不等式，以及适用于中心极限定理中熵产生研究的该定理的局部版本。
The Brunn�Minkowski inequality gives a lower bound of the Lebesgue measure of a sum-set in terms of the measures of the individual sets. It has played a crucial role in the theory of convex bodies. This topic has many interactions with isoperimetry or functional analysis. Our aim here is to report some recent aspects of these interactions involving optimal mass transport or the Heat equation. Among other things, we will present Brunn�Minkowski inequalities for flat sets, or in Gauss space, as well as local versions of the theorem which apply to the study of entropy production in the central limit theorem.