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 a sharp mixing time bound for the bases-exchange walk for matroids, and a concentration bound for Lipschitz functions over these distributions.
"Modified log-Sobolev inequalities for strongly log-concave distributions." Ann. Probab. 49 (1) 506 - 525, January 2021. https://doi.org/10.1214/20-AOP1453