On Sharp Sobolev Embedding and The Logarithmic Sobolev Inequality

The purpose of this note is to give a short proof of the Gross logarithmic Sobolev inequality using the asymptotics of the sharp L 2 Sobolev constant and the product structure of Euclidean space. Let F ∈ L r (IR n ) for some positive r with ∥ F ∥ r =1.