Logarithmic Sobolev Inequalities and the Information Theory
نویسندگان
چکیده
In this paper we present an overview on logarithmic Sobolev inequalities. These inequalities have become a subject of intense research activity during the past years, from analysis and geometry in finite and infinite dimension, to probability and statistical mechanics, and of course many others developments and applications are expected. We have divided this paper into three parts. The first part includes the initial forms of Poincaré inequalities and Logarithmic Sobolev Inequalities for the Bernoulli and Gauss measure. In the second part the relationships between logarithmic Sobolev inequalities and the transportation of measures are considered, useful in statistics as well as geometry. The last part is a modern reading of entropy in information theory and of several links between information theory and the Euclidean form of the logarithmic Sobolev inequality. The genesis and the introduction of these inequalities can be traced back in the pioneering work of Shannon and Stam.
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