Title: Shannon's problem on the monotonicity of entropy.

Abstract: The entropy of a random variable with density f is defined as
the integral of  -f log f. In the 1940s Shannon proved that if X_1 and
X_2 are i.i.d., then the entropy of (X_1+X_2)/sqrt{2} is at least the
entropy of X_1. The problem whether, for i.i.d. random variables X_i,
the entropy of (X_1+...+X_n)/sqrt{n} is an increasing sequence remained
open. In this talk we will show that, indeed, entropy increases along
central limit averages. The proof is based on a new variational formula
which is motivated by (a proof of) the Brunn-Minkowski inequality.

Based on joint works with S. Artstein, K. Ball and F. Barthe.