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 \(h(f)\ge m+\tfrac12\log\det\mathop{\mathrm{Cov}}\nolimits (f)\) for every log-concave probability density on ℝm, with equality precisely for invertible affine images of products of one-sided exponential laws.
Cite
@misc{OAI:A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026,
author = {{OpenAI}},
title = {{A sharp entropy bound and the simplex inequality for isotropic constants}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/isotropic-simplex.pdf}{OAI:A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026}},
year = {2026}
}