The ball as the least densely packable convex solid
Let $K \subset \mathbb{R}^3$ be a convex body, that is, a compact convex set with nonempty interior. A packing of $K$ is a locally finite family $\mathcal{P} = \{a_i + Q_i K : i \in I\}$, with $a_i \in \mathbb{R}^3$ and $Q_i \in O(3)$, whose members have pairwise disjoint interiors; rotated and reflected copies are allowed, and the packing need not be periodic. With $B_R$ the ball of radius $R$ centred at the origin, let $$\overline{d}(\mathcal{P}) = \limsup_{R \to \infty} \frac{\bigl|B_R \cap \bigcup_{P \in \mathcal{P}} P\bigr|}{|B_R|}, \qquad \delta(K) = \sup_{\mathcal{P}} \overline{d}(\mathcal{P}),$$ the supremum over all packings of $K$. For a ball $B$, Hales's proof of the Kepler conjecture gives $\delta(B) = \pi/\sqrt{18}$. Kallus proved that every origin-symmetric convex body close enough to a ball, other than a ball, has $\delta(K) > \pi/\sqrt{18}$, and that the double-lattice packing density, a lower bound for $\delta$, increases at a positive rate along every path $(1 - \lambda) B + \lambda K$, $\lambda \downarrow 0$, from the ball towards a body $K$ in minimal mean width position that is not a ball. Regular tetrahedra, once suggested as a counterexample, pack with density $4000/4671 > \pi/\sqrt{18}$ (Chen, Engel and Glotzer).
Problem. Is $\delta(K) \ge \pi/\sqrt{18}$ for every convex body $K \subset \mathbb{R}^3$? No symmetry or smoothness of $K$ is assumed.
Source: Ulam (1972), in a remark to Gardner recorded in the postscript to "Packing spheres" in Gardner, New Mathematical Diversions (revised edition, 1995); named Ulam's packing conjecture by Kallus (2014) and stated by Kallus (2015).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- CO
- Keywords
- AIM Open Applied Problems, convex bodies, Kepler conjecture, packing densities
References
@book{gardner1995,
author = {Gardner, Martin},
title = {New Mathematical Diversions},
edition = {Revised},
publisher = {Mathematical Association of America},
address = {Washington, DC},
year = {1995},
note = {Postscript to the chapter ``Packing spheres''},
}
@article{kallus2014,
author = {Kallus, Yoav},
title = {The 3-ball is a local pessimum for packing},
journal = {Adv. Math.},
volume = {264},
year = {2014},
pages = {355--370},
doi = {10.1016/j.aim.2014.07.015},
eprint = {1212.2551},
archivePrefix = {arXiv},
}
@article{kallus2015,
author = {Kallus, Yoav},
title = {Pessimal packing shapes},
journal = {Geom. Topol.},
volume = {19},
number = {1},
year = {2015},
pages = {343--363},
doi = {10.2140/gt.2015.19.343},
eprint = {1305.0289},
archivePrefix = {arXiv},
}
@article{hales2005,
author = {Hales, Thomas C.},
title = {A proof of the {K}epler conjecture},
journal = {Ann. of Math. (2)},
volume = {162},
number = {3},
year = {2005},
pages = {1065--1185},
doi = {10.4007/annals.2005.162.1065},
}
@article{chen2010,
author = {Chen, Elizabeth R. and Engel, Michael and Glotzer, Sharon C.},
title = {Dense crystalline dimer packings of regular tetrahedra},
journal = {Discrete Comput. Geom.},
volume = {44},
number = {2},
year = {2010},
pages = {253--280},
doi = {10.1007/s00454-010-9273-0},
eprint = {1001.0586},
archivePrefix = {arXiv},
}
@misc{aim-ulam-convex-solid-packing,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The ball as the least densely packable convex solid}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/349-ulam-convex-solid-packing.md},
note = {Entry 349 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00653, "The ball as the least densely packable convex solid", clopen.solutions.