Maximum convex-hull volume for a closed curve of fixed length
Let $\gamma\colon [0,1] \to \mathbb{R}^3$ range over continuous rectifiable curves with $\gamma(0) = \gamma(1)$ and $\operatorname{Length}(\gamma) = 1$. Self-intersections are allowed, and no symmetry or bound on the number of intersections with a plane is assumed. Melzak studied the problem for closed curves under a symmetry constraint. Bohr, Markvorsen and Raffaelli gave an upper bound for the volume of the convex hull of a simple closed Frenet curve with exactly four vertices (points of vanishing torsion) lying on the boundary of its convex hull, and showed that it is attained when the curve meets every plane in at most four points.
Problem. Determine $$V_* = \sup_\gamma \operatorname{Vol}_3\bigl(\operatorname{conv}(\gamma([0,1]))\bigr)$$ and characterize the maximizing curves up to rigid motion and reparametrization.
Source: Bonnesen and Fenchel (1934), p. 117; see Croft, Falconer and Guy (1991), A28, and Bohr, Markvorsen and Raffaelli (2026), Problem 1.1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, convex hulls, isoperimetric problems, space curves
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},
}
@book{croft1991,
author = {Croft, Hallard T. and Falconer, Kenneth J. and Guy, Richard K.},
title = {Unsolved Problems in Geometry},
series = {Problem Books in Mathematics},
publisher = {Springer},
address = {New York},
year = {1991},
doi = {10.1007/978-1-4612-0963-8},
}
@article{bohr2026,
author = {Bohr, Jakob and Markvorsen, Steen and Raffaelli, Matteo},
title = {The convex hull of a convex space curve with four vertices},
journal = {Bull. Lond. Math. Soc.},
volume = {58},
number = {3},
year = {2026},
pages = {e70311},
doi = {10.1112/blms.70311},
eprint = {1805.11335},
archivePrefix = {arXiv},
}
@article{melzak1960,
author = {Melzak, Z. A.},
title = {The isoperimetric problem of the convex hull of a closed space curve},
journal = {Proc. Amer. Math. Soc.},
volume = {11},
number = {2},
year = {1960},
pages = {265--274},
doi = {10.1090/S0002-9939-1960-0116263-0},
}
@misc{aim-closed-curve-hull-volume,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Maximum convex-hull volume for a closed curve of fixed length}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/247-closed-curve-hull-volume.md},
note = {Entry 247 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-00510, "Maximum convex-hull volume for a closed curve of fixed length", clopen.solutions.