An information-theoretic proof of Nash's inequality

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. The research that led to the present paper was partially supported by a grant of the group GNFM of INdAM.