A sharp entropy bound and the simplex inequality for isotropic constants

We prove the strong isotropic constant conjecture: in each dimension, simplices are the unique maximizers of the isotropic constant among convex bodies. We also prove the sharp entropy bound $h(f)\ge m+\frac{1}{2}\log \det \mathrm{Cov}(f)$ for every log-concave probability density on ℝ${}^m$, with equality precisely for invertible affine images of products of one-sided exponential laws.