Bizeul's conjecture: a dimension-free log-Sobolev inequality for log-concave measures with subgaussian linear marginals

Theorem 1.1: there is an absolute $C$ such that for every $n$, every centered log-concave probability $\mu$ on $\mathbb R^n$ with a density and linear subgaussian parameter $a$, $\mathrm{Ent}_\mu(f^2)\le Ca^2\int|Df|^2d\mu$ for all $f\in C_c^\infty$; this includes uniform measures on convex bodies and needs no smoothness or curvature. By Otto-Villani it gives the transport-entropy inequality $W_2(\nu,\mu)^2\le Ca^2H(\nu|\mu)$ with the same constant. Not given: a value of $C$, or anything for non-log-concave measures.

Theorem 1.1: there is an absolute $C$ such that for every $n$, every centered log-concave probability $\mu$ on $\mathbb R^n$ with a density and linear subgaussian parameter $a$, $\mathrm{Ent}_\mu(f^2)\le Ca^2\int|Df|^2d\mu$ for all $f\in C_c^\infty$; this includes uniform measures on convex bodies and needs no smoothness or curvature. By Otto-Villani it gives the transport-entropy inequality $W_2(\nu,\mu)^2\le Ca^2H(\nu|\mu)$ with the same constant. Not given: a value of $C$, or anything for non-log-concave measures.