The shortest space curve of prescribed minimum width
For a continuous rectifiable curve $\gamma\colon [0,1] \to \mathbb{R}^3$ let $$w(\gamma) = \min_{u \in S^2} \Bigl( \max_{t \in [0,1]} \gamma(t) \cdot u - \min_{t \in [0,1]} \gamma(t) \cdot u \Bigr),$$ the least distance between two parallel planes that enclose its image. The endpoints are free and the curve need not be closed. Zalgaller proposed a curve $L_3$, made of segments and helical arcs, of length about $3.9215$, and conjectured that it is the shortest curve of width one. Bartels, Choy, Wu and Li showed in a 2026 preprint that $L_3$ has width less than $0.9912$, and constructed, with a computer-assisted proof formalized in Lean, a curve of width one and length $3.905124\ldots$, which they prove is a strict local minimum within a candidate family. Ghomi proved that $\operatorname{Length}(\gamma) \ge 3.7669\, w(\gamma)$ for every such curve. For closed curves he proved $\operatorname{Length}(\gamma) \ge \sqrt{\pi^2 + 16}\, w(\gamma)$ and constructed a closed curve of width one and length less than $5.1151$, shorter than Zalgaller's conjectured closed minimizer.
Problem. Determine the exact infimum of $\operatorname{Length}(\gamma)$ subject to $w(\gamma) = 1$, and identify all minimizing images up to rigid motion.
Source: Zalgaller (1994), Conjecture 1, that his curve L₃ is the shortest space curve of width 1; a space analogue of Bellman's escape-path problem (Ghomi, 2018).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, escape paths, space curves, width
References
@article{zalgaller1994,
author = {Zalgaller, V. A.},
title = {The shortest space curve of unit width},
journal = {Mat. Fiz. Anal. Geom.},
volume = {1},
number = {3-4},
year = {1994},
pages = {454--461},
language = {Russian},
note = {English translation by S. Finch},
eprint = {1910.02729},
archivePrefix = {arXiv},
}
@article{ghomi2018,
author = {Ghomi, Mohammad},
title = {The length, width, and inradius of space curves},
journal = {Geom. Dedicata},
volume = {196},
number = {1},
year = {2018},
pages = {123--143},
doi = {10.1007/s10711-017-0312-3},
eprint = {1605.01144},
archivePrefix = {arXiv},
}
@misc{bartels2026,
author = {Bartels, Tinko Sebastian and Choy, Dennis Kevin and Wu, Ruixiang and Li, Tongxin},
title = {A short open space curve of unit width},
year = {2026},
eprint = {2610.19585},
archivePrefix = {arXiv},
}
@misc{aim-shortest-width-one-space-curve,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The shortest space curve of prescribed minimum width}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/246-shortest-width-one-space-curve.md},
note = {Entry 246 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-00555, "The shortest space curve of prescribed minimum width", clopen.solutions.