Exact minimum length of an opaque barrier for a square
Let $Q = [0,1]^2$. A rectifiable barrier for $Q$ is a set $B \subset \mathbb{R}^2$ that is a countable union of rectifiable curves and meets every straight line that meets $Q$; it need not be connected or contained in $Q$. Let $$b(Q) = \inf\{\mathcal{H}^1(B) : B \text{ is a rectifiable barrier for } Q\},$$ where $\mathcal{H}^1$ is one-dimensional Hausdorff measure. Kawamura, Moriyama, Otachi and Pach showed that every rectifiable barrier can be replaced by a union of countably many line segments of length arbitrarily close to its own. Jones proved that a rectifiable barrier of a compact convex set of perimeter $2p$ has length at least $p$, so $b(Q) \ge 2$; Kawamura, Moriyama, Otachi and Pach raised the bound to $2.00002$, and Kiderlen and Pausinger to $2 + 6.3 \cdot 10^{-5}$. For barriers contained in $Q$, Dumitrescu and Jiang proved the bound $2 + 10^{-5}$. Jones found a barrier of length $\sqrt{2} + \sqrt{6}/2 = 2.6389\ldots$ (Jones), consists of half a diagonal, from the centre to a corner, and the three segments joining the other three corners to their Steiner point.
Problem. Determine $b(Q)$ exactly. The infimum is not assumed to be attained.
Source: Mazurkiewicz (1916), who first considered sets meeting every line through a square, and Bagemihl (1959), who named them opaque sets; for rectifiable barriers, Kawamura, Moriyama, Otachi and Pach (2019) and Kiderlen and Pausinger (2026), §1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- CO
- Keywords
- AIM Open Applied Problems, barriers, opaque sets
References
@article{mazurkiewicz1916,
author = {Mazurkiewicz, Stefan},
title = {Przyk{\l}ad zbioru domkni{\k{e}}tego, punktokszta{\l}tnego, maj{\k{a}}cego punkty wsp{\'o}lne z ka{\.z}d{\k{a}} prost{\k{a}}, przecinaj{\k{a}}c{\k{a}} pewien obszar domkni{\k{e}}ty},
journal = {Prace Mat.-Fiz.},
volume = {27},
year = {1916},
pages = {11--16},
note = {French summary: Sur un ensemble ferm{\'e}, punctiforme, qui rencontre toute droite passant par un certain domaine},
}
@article{bagemihl1959,
author = {Bagemihl, F.},
title = {Some opaque subsets of a square},
journal = {Michigan Math. J.},
volume = {6},
number = {2},
year = {1959},
pages = {99--103},
doi = {10.1307/mmj/1028998183},
}
@article{kawamura2019,
author = {Kawamura, Akitoshi and Moriyama, Sonoko and Otachi, Yota and Pach, J{\'a}nos},
title = {A lower bound on opaque sets},
journal = {Comput. Geom.},
volume = {80},
year = {2019},
pages = {13--22},
doi = {10.1016/j.comgeo.2019.01.002},
eprint = {1403.3894},
archivePrefix = {arXiv},
}
@article{kiderlen2026,
author = {Kiderlen, Markus and Pausinger, Florian},
title = {Explicit lower bounds for opaque sets of unit square and unit disc},
journal = {Acta Math. Hungar.},
volume = {178},
number = {1},
year = {2026},
pages = {289--303},
doi = {10.1007/s10474-026-01606-x},
eprint = {2509.08842},
archivePrefix = {arXiv},
}
@article{jones1964,
author = {Jones, R. E. D.},
title = {Opaque sets of degree $\alpha$},
journal = {Amer. Math. Monthly},
volume = {71},
number = {5},
year = {1964},
pages = {535--537},
doi = {10.2307/2312596},
}
@phdthesis{jones1962,
author = {Jones, R. E. D.},
title = {Linear Measure and Opaque Sets},
school = {Iowa State University},
year = {1962},
doi = {10.31274/rtd-180815-2223},
}
@inproceedings{dumitrescu2014,
author = {Dumitrescu, Adrian and Jiang, Minghui},
title = {The opaque square},
booktitle = {Proceedings of the Thirtieth Annual Symposium on Computational Geometry (SoCG '14)},
publisher = {ACM},
address = {New York},
year = {2014},
pages = {529--538},
doi = {10.1145/2582112.2582113},
eprint = {1311.3323},
archivePrefix = {arXiv},
}
@misc{aim-shortest-opaque-square-barrier,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Exact minimum length of an opaque barrier for a square}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/122-shortest-opaque-square-barrier.md},
note = {Entry 122 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-00554, "Exact minimum length of an opaque barrier for a square", clopen.solutions.