Asymmetry at the Macbeath point of a convex body
Let $K \subset \mathbb{R}^d$ be compact and convex with nonempty interior. Macbeath showed that the function $$x \longmapsto \operatorname{vol}_d\bigl(K \cap (2x - K)\bigr), \qquad 2x - K = \{2x - y : y \in K\},$$ attains its maximum at a unique point $p$ of the interior of $K$, the Macbeath point of $K$. Ivanov observes that quantitative Carathéodory-type and Helly-type results require a center deep inside the body, such as the center of the John ellipsoid, the barycenter or the Santaló point, and believes that the Macbeath point might open a new direction for volumetric Helly-type results and coarse approximations of convex bodies.
Problem. Does every compact convex $K \subset \mathbb{R}^d$ with nonempty interior satisfy $$K - p \subseteq -d\,(K - p),$$ where $p$ is its Macbeath point? Equivalently: is $p - (y - p)/d \in K$ for every $y \in K$? The center is the Macbeath point; another center with this property does not answer the question.
Source: Ivanov (2025), Conjecture 2.1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MG — Metric Geometry
- Keywords
- AIM Open Applied Problems, centers of convex bodies, convex bodies
References
@article{ivanov2026,
author = {Ivanov, Grigory},
title = {Quantitative {S}teinitz theorem and polarity},
journal = {Discrete Comput. Geom.},
volume = {75},
number = {4},
year = {2026},
pages = {1405--1413},
doi = {10.1007/s00454-025-00743-4},
eprint = {2403.14761},
archivePrefix = {arXiv},
}
@article{macbeath1952,
author = {Macbeath, A. M.},
title = {A theorem on non-homogeneous lattices},
journal = {Ann. of Math. (2)},
volume = {56},
number = {2},
year = {1952},
pages = {269--293},
doi = {10.2307/1969800},
}
@misc{aim-macbeath-center-asymmetry,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Asymmetry at the Macbeath point of a convex body}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/565-macbeath-center-asymmetry.md},
note = {Entry 565 at commit 24edaad, 2026-10-15},
}
Resolution claims
The center of a largest centrally symmetric subset of a convex body need not satisfy the usual dimension bound on asymmetry. We compute this center for a square pyramid in $\mathbb R^3$ and show that reflection about it, followed by shrinking by a factor of three, takes the apex outside the pyramid. This gives a counterexample to Ivanov's Macbeath point …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/macbeath_counterexample.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00808, "Asymmetry at the Macbeath point of a convex body", clopen.solutions.