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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In A sharp entropy bound and the simplex inequality for isotropic constants, 5 October 2026
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For every integer n\ge1, every convex body K\subset\R^n satisfies the following bound, with equality if and only if K is a simplex:
In A sharp entropy bound and the simplex inequality for isotropic constants, 5 October 2026
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Let m\ge1 and let f be a log-concave probability density on \R^m. Then
Equality holds if and only if f is, almost everywhere, the density of M(E_1,\ldots,E_m)^{\mathsf T}+b, where M is an invertible real m\times m matrix, b\in\R^m, and the E_i are independent random variables with density e^{-u}\mathbf1_{\{u\ge0\}}.
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Lemmas 4.1 and 2.2, and Section 4 of A sharp entropy bound and the simplex inequality for isotropic constants, 5 October 2026
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Why this passage is shown
Theorem 1.1 (The simplex inequality)
In A sharp entropy bound and the simplex inequality for isotropic constants, 5 October 2026
Why this passage is shown
Theorem 1.2 (The sharp entropy bound)
In A sharp entropy bound and the simplex inequality for isotropic constants, 5 October 2026
Report using another proof or calculation method
Lemmas 4.1 and 2.2, and Section 4 of A sharp entropy bound and the simplex inequality for isotropic constants, 5 October 2026
Exact calculations for finite claims, cert-machine
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