The sharp simplex conjecture for isotropic constants
OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
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.
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MANUSCRIPT 01
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)≥ m+\tfrac12log\det\mathop{Cov}\nolimits (f) for every log-concave probability density on ℝ^(m), with equality precisely for…
Full source abstract
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+\tfrac12log\det\mathop{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.
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