An information-theoretic proof of Nash's inequality
Summary: We show that an information-theoretic property of Shannon's entropy power, known as concavity of entropy power [\textit{M. H. M. Costa}, IEEE Trans. Inf. Theory 31, 751--760 (1985; Zbl 0585.94006)], can be fruitfully employed to prove inequalities in sharp form. In particular, the concavity of entropy power implies the logarithmic Sobolev inequality, and the Nash's inequality with the sharp constant.
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