A sharp surface-area bound for the convex hull of a closed curve
Let $\gamma : [0,1] \to \mathbb{R}^3$ be a continuous rectifiable closed curve of length $L$, and let $K = \operatorname{conv}(\gamma([0,1]))$. Let $A(K)$ be the surface area of $\partial K$ when $K$ is three-dimensional, twice the area of $K$ when $K$ is two-dimensional, and zero otherwise. For a planar circle, $K$ is a doubly covered disk and $A(K) = L^2/(2\pi)$. As Ghomi notes, when $\gamma$ is simple and lies on $\partial K$, it divides $\partial K$ into two disks of zero curvature, and Weil's isoperimetric inequality for surfaces of nonpositive curvature then gives $A(K) \le L^2/(2\pi)$.
Problem. Is $$A(K) \le \frac{L^2}{2\pi}$$ for every such curve $\gamma$? The curve need not lie on the boundary of its convex hull.
Source: Croft, Falconer and Guy (1991), p. 38, as traced by Ghomi (MathOverflow question 480430, 2024); see Ghomi (2019), Problem 5.3, who asks to show that the area of the convex hull of a closed space curve of fixed length is largest for a circle.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, convex hulls, isoperimetric inequalities, space curves
References
@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-Verlag},
address = {New York},
year = {1991},
doi = {10.1007/978-1-4612-0963-8},
}
@misc{ghomi2019,
author = {Ghomi, Mohammad},
title = {Open problems in geometry of curves and surfaces},
year = {2019},
url = {https://ghomi.math.gatech.edu/Papers/op.pdf},
note = {Last revised September 2, 2019},
}
@article{weil1926,
author = {Weil, Andr{\'e}},
title = {Sur les surfaces {\`a} courbure n{\'e}gative},
journal = {C. R. Acad. Sci. Paris},
volume = {182},
year = {1926},
pages = {1069--1071},
}
@misc{aim-closed-curve-hull-area,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A sharp surface-area bound for the convex hull of a closed curve}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/248-closed-curve-hull-area.md},
note = {Entry 248 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-00509, "A sharp surface-area bound for the convex hull of a closed curve", clopen.solutions.