We show that an information-theoretic property of Shannon’s entropy power, known as concavity of entropy power , 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.
Cite this article
Giuseppe Toscani, An information-theoretic proof of Nash’s inequality. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 24 (2013), no. 1, pp. 83–93DOI 10.4171/RLM/645