Minimum volume of a three-dimensional body of constant width
For a convex body $K \subset \mathbb{R}^3$ with nonempty interior let $h_K(u) = \max_{x \in K} x \cdot u$, $u \in S^2$, be its support function; $K$ has constant width one if $h_K(u) + h_K(-u) = 1$ for every $u \in S^2$. In the plane, the Reuleaux triangle has the least area among convex bodies of given constant width (Lebesgue, 1914; Blaschke, 1915). In three dimensions, Bonnesen and Fenchel conjectured that the minimizers are Meissner's bodies, two non-congruent bodies of constant width with the same volume, about $0.419860$ for width one. Chakerian proved that every body of constant width one has volume at least $\frac{\pi}{3}(3\sqrt{6} - 7) \approx 0.364916$, and Nishioka improved this to $\frac{4\pi}{33} \approx 0.380799$ (2026), and Jiang to $\frac{1651}{4000} = 0.41275$ (2026). Meissner polyhedra are dense among bodies of constant width in the Hausdorff metric (Hynd, essentially already Sallee).
Problem. Determine the exact value of $$v_3 = \inf\bigl\{\operatorname{Vol}_3(K) : K \subset \mathbb{R}^3 \text{ a convex body of constant width one}\bigr\},$$ and characterize, up to rigid motion, all bodies attaining it. No rotational or polyhedral symmetry is imposed.
Source: Bonnesen and Fenchel (1934), who conjectured that Meissner's bodies have the least volume among three-dimensional bodies of constant width; see Kawohl and Weber (2011). Stated by AIM — Open Applied Problems as the determination of the minimum and of all minimizers.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, Blaschke–Lebesgue problem, bodies of constant width, Meissner bodies
References
@book{bonnesen1934,
author = {Bonnesen, T. and Fenchel, W.},
title = {Theorie der konvexen {K}{\"o}rper},
series = {Ergebnisse der Mathematik und ihrer Grenzgebiete},
volume = {3},
publisher = {Springer},
address = {Berlin},
year = {1934},
doi = {10.1007/978-3-642-47404-0},
}
@article{kawohl2011,
author = {Kawohl, Bernd and Weber, Christof},
title = {Meissner's mysterious bodies},
journal = {Math. Intelligencer},
volume = {33},
number = {3},
year = {2011},
pages = {94--101},
doi = {10.1007/s00283-011-9239-y},
}
@article{lebesgue1914,
author = {Lebesgue, Henri},
title = {Sur le probl{\`e}me des isop{\'e}rim{\`e}tres et sur les domaines de largeur constante},
journal = {Bull. Soc. Math. France},
volume = {42},
year = {1914},
pages = {72--76},
note = {Comptes rendus des s{\'e}ances},
}
@article{blaschke1915,
author = {Blaschke, Wilhelm},
title = {Konvexe {B}ereiche gegebener konstanter {B}reite und kleinsten {I}nhalts},
journal = {Math. Ann.},
volume = {76},
number = {4},
year = {1915},
pages = {504--513},
doi = {10.1007/BF01458221},
}
@article{chakerian1966,
author = {Chakerian, G. D.},
title = {Sets of constant width},
journal = {Pacific J. Math.},
volume = {19},
number = {1},
year = {1966},
pages = {13--21},
doi = {10.2140/pjm.1966.19.13},
}
@misc{nishioka2026,
author = {Nishioka, Akatsuki},
title = {An improved lower bound for the three-dimensional {B}laschke--{L}ebesgue problem from spectral and dual perspectives},
year = {2026},
eprint = {2606.01754},
archivePrefix = {arXiv},
}
@misc{jiang2026,
author = {Jiang, Yunjiang},
title = {A 0.41275 lower bound for the three-dimensional {B}laschke--{L}ebesgue problem},
year = {2026},
eprint = {2610.13964},
archivePrefix = {arXiv},
}
@article{hynd2024,
author = {Hynd, Ryan},
title = {The density of {M}eissner polyhedra},
journal = {Geom. Dedicata},
volume = {218},
number = {4},
year = {2024},
pages = {89},
doi = {10.1007/s10711-024-00933-z},
eprint = {2304.04035},
archivePrefix = {arXiv},
}
@misc{aim-constant-width-volume,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Minimum volume of a three-dimensional body of constant width}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/242-constant-width-volume.md},
note = {Entry 242 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00511, "Minimum volume of a three-dimensional body of constant width", clopen.solutions.