The sharp simplex conjecture for isotropic constants

Proves that simplices uniquely maximize the isotropic constant among convex bodies in every dimension, resolving the strong isotropic constant conjecture. Also establishes the sharp entropy lower bound for log-concave probability densities, with equality precisely for invertible affine images of products of one-sided exponential laws.

Manuscript

A sharp entropy bound and the simplex inequality for isotropic constants

October 5, 2026 · source · cite as \cite{OAI:A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026}

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\mathop{\mathrm{Cov}}\nolimits (f) for every log-concave probability density on ℝ^m, with equality precisely for invertible affine images of products of one-sided exponential laws.

Result 101 · Convex and metric geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.