Simplices maximize the isotropic constant
Let $K \subset \mathbb{R}^n$ be a compact convex set with nonempty interior, let $X$ be uniformly distributed on $K$, and put $$c_K = \mathbb{E} X, \qquad A_K = \mathbb{E}\bigl[(X - c_K)(X - c_K)^{\mathsf T}\bigr], \qquad L_K = \left(\frac{\det A_K}{\operatorname{vol}_n(K)^2}\right)^{1/(2n)}.$$ The isotropic constant $L_K$ is invariant under invertible affine maps. For a nondegenerate $n$-simplex $\Delta_n$, $$L_{\Delta_n} = \frac{(n!)^{1/n}}{(n+1)^{(n+1)/(2n)} \sqrt{n+2}}.$$ Klartag and Lehec proved that $L_K$ is bounded above by a universal constant (Bourgain's slicing problem). In the plane the inequality below follows from a result of Campi, Colesanti and Gronchi, and Rademacher proved that a simplicial polytope maximizing the isotropic constant must be a simplex.
Problem. Is $L_K \le L_{\Delta_n}$ for every $n \ge 1$ and every such $K$? Only the inequality is asked, not uniqueness or stability of the maximizers.
Source: The strong isotropic constant conjecture, as stated by Kipp (2026), Section 1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- FA, PR
- Keywords
- AIM Open Applied Problems, convex bodies, isotropic constant, slicing problem
References
@article{kipp2026,
author = {Kipp, Christian},
title = {Shadow systems, decomposability and isotropic constants},
journal = {Discrete Comput. Geom.},
volume = {76},
number = {3},
year = {2026},
pages = {1391--1425},
doi = {10.1007/s00454-026-00829-7},
eprint = {2411.08722},
archivePrefix = {arXiv},
}
@article{klartag2025,
author = {Klartag, Boaz and Lehec, Joseph},
title = {Affirmative resolution of {B}ourgain's slicing problem using {G}uan's bound},
journal = {Geom. Funct. Anal.},
volume = {35},
number = {4},
year = {2025},
pages = {1147--1168},
doi = {10.1007/s00039-025-00718-w},
eprint = {2412.15044},
archivePrefix = {arXiv},
}
@article{campi1999,
author = {Campi, Stefano and Colesanti, Andrea and Gronchi, Paolo},
title = {A note on {S}ylvester's problem for random polytopes in a convex body},
journal = {Rend. Ist. Mat. Univ. Trieste},
volume = {31},
year = {1999},
pages = {79--94},
}
@article{rademacher2016,
author = {Rademacher, Luis},
title = {A simplicial polytope that maximizes the isotropic constant must be a simplex},
journal = {Mathematika},
volume = {62},
number = {1},
year = {2016},
pages = {307--320},
doi = {10.1112/S0025579315000133},
eprint = {1404.5662},
archivePrefix = {arXiv},
}
@misc{aim-simplex-maximal-isotropic-constant,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Simplices maximize the isotropic constant}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/543-simplex-maximal-isotropic-constant.md},
note = {Entry 543 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
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 $h(f)\ge m+\tfrac12\log\det\Cov(f)$ for every log-concave probability density on $\R^m$, with equality precisely for invertible affine images of products of one-sided exponential laws.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, manuscript/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00716, "Simplices maximize the isotropic constant", clopen.solutions.