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 Nash's inequality with the sharp constant. 
Giuseppe Toscani
cs.IT math-ph math.FA math.IT math.MP