Given a variance one random variable on the line it is an elementary and classical fact that the Gaussian has maximal Shannon entropy. In this talk we show that the exponential distribution has minimal entropy among fixed variance log-concave random variables. This answers a 2010 question of Bobkov and Madiman and shows that the exponential distribution has maximal isotropic constant (in its entropic form) among log-concave random variables on the line. Our proof methods use the degrees of freedom approach of Fradelizi and Guedon as well as a rearrangement inequality for unimodal densities. We will also provide a partial solution to the Renyi entropy generalization. This is joint work with Piotr Nayar and Cyril Roberto.
