The sharp simplex conjecture for isotropic constants

What OpenAI claims

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.

Paper

1. A sharp entropy bound and the simplex inequality for isotropic constants

   5 October 2026 · Preprint, not peer reviewed · PDF

   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) ≥ m+1/2 log det Cov (f) for every log-concave probability density on ℝm, with equality precisely for invertible affine images of products of one-sided exponential laws.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.