A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures

We prove that every centered log-concave probability measure with a Lebesgue density on ℝ${}^n$ and linear subgaussian parameter a satisfies ${\mathrm{Ent}}_{\mu }(f^2)\le C{a}^2\int |Df{|}^{2}d\mu$ for compactly supported smooth f, with one universal constant C. This resolves positively the dimension-free logarithmic Sobolev conjecture for subgaussian log-concave measures.