The strong isotropic constant conjecture: simplices maximize the isotropic constant

Claims Theorem 1.1: for every $n\ge1$ and every convex body $K\subset\mathbb R^n$, $L_K\le(n!)^{1/n}/((n+1)^{(n+1)/(2n)}\sqrt{n+2})$, with equality if and only if $K$ is a simplex. Also claims the sharp entropy bound $h(f)\ge m+\tfrac12\log\det\mathrm{Cov}(f)$ for every log-concave density on $\mathbb R^m$, with equality exactly for affine images of products of one-sided exponentials, proved via Gaussian-to-target Brenier transport. Via Klartag (2018) it yields the non-symmetric Mahler inequality $|K||K^\circ|\ge(n+1)^{n+1}/(n!)^2$, already listed from a separate manuscript of the same release. It gives no new information on the symmetric slicing constant or on symmetric Mahler.

Claims Theorem 1.1: for every $n\ge1$ and every convex body $K\subset\mathbb R^n$, $L_K\le(n!)^{1/n}/((n+1)^{(n+1)/(2n)}\sqrt{n+2})$, with equality if and only if $K$ is a simplex. Also claims the sharp entropy bound $h(f)\ge m+\tfrac12\log\det\mathrm{Cov}(f)$ for every log-concave density on $\mathbb R^m$, with equality exactly for affine images of products of one-sided exponentials, proved via Gaussian-to-target Brenier transport. Via Klartag (2018) it yields the non-symmetric Mahler inequality $|K||K^\circ|\ge(n+1)^{n+1}/(n!)^2$, already listed from a separate manuscript of the same release. It gives no new information on the symmetric slicing constant or on symmetric Mahler.