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 $h(f)\ge m+\tfrac12\log\det\Cov(f)$ for every log-concave probability density on $\R^m$, with equality precisely for invertible affine images of products of one-sided exponential laws.