Modified log-Sobolev Inequalities for Strongly Log-Concave Distributions

We show that the modified log-Sobolev constant for a natural Markov chain which converges to an r-homogeneous strongly log-concave distribution is at least 1/r. Applications include an asymptotically optimal mixing time bound for the bases-exchange walk for matroids, and a concentration bound for Lipschitz functions over these distributions.