An information-theoretic proof of Nash's inequality

TL;DR

This paper uses an information-theoretic approach based on Shannon's entropy power to provide a sharp proof of Nash's inequality, connecting entropy properties with functional inequalities.

Contribution

It introduces a novel proof technique leveraging entropy power concavity to establish Nash's inequality with the optimal constant.

Findings

Proves Nash's inequality using entropy power concavity

Establishes a connection between entropy properties and functional inequalities

Provides a sharp constant proof for Nash's inequality

Abstract

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 Nash's inequality with the sharp constant.