The sharp number of directions illuminating a convex body
Let $d \ge 3$ and let $K \subset \mathbb{R}^d$ be a convex body. A direction $v \in S^{d-1}$ illuminates a boundary point $x \in \partial K$ if $x + tv \in \operatorname{int} K$ for some $t > 0$, and the illumination number $I(K)$ is the least number of directions that together illuminate every point of $\partial K$. Boltyanski proved that $I(K)$ is also the least number of translates of $\operatorname{int} K$ that cover $K$, equivalently the least number of smaller positive homothetic copies $a + \lambda K$, $0 < \lambda < 1$, that cover $K$. An affine cube needs $2^d$ directions, one for each vertex. Levi settled the planar case. In dimension three, Lassak proved $I(K) \le 8$ for centrally symmetric bodies, and Prymak proved $I(K) \le 14$ for every convex body. Campos, van Hintum, Morris and Tiba proved $I(K) \le e^{-cd/L_K^2}\, 4^d$ for all sufficiently large $d$, where $L_K$ is the isotropic constant of $K$ and $c > 0$ is universal, so that the affirmative answer to Bourgain's slicing problem, given by Klartag and Lehec, yields a bound $e^{-c'd}\, 4^d$.
Problem. Is $I(K) \le 2^d$ for every $d \ge 3$ and every convex body $K \subset \mathbb{R}^d$, with $I(K) = 2^d$ if and only if $K = a + A[0,1]^d$ for some $a \in \mathbb{R}^d$ and some invertible linear map $A$?
Source: Hadwiger (1957), as a covering problem, after Levi's planar case (1955); posed independently by Gohberg and Markus (1960), and as an illumination problem by Boltyanski (1960) and Hadwiger (1960). In this form, Bezdek and Khan (2018), Conjecture 1.2.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- CO
- Keywords
- AIM Open Applied Problems, convex bodies, homothetic coverings, illumination numbers
References
@article{hadwiger1957,
author = {Hadwiger, Hugo},
title = {Ungel{\"o}ste {P}robleme {N}r. 20},
journal = {Elem. Math.},
volume = {12},
year = {1957},
pages = {121},
}
@article{levi1955,
author = {Levi, F. W.},
title = {{\"U}berdeckung eines {E}ibereiches durch {P}arallelverschiebung seines offenen {K}erns},
journal = {Arch. Math.},
volume = {6},
number = {5},
year = {1955},
pages = {369--370},
doi = {10.1007/BF01900507},
}
@article{gohberg1960,
author = {Gohberg, I. Ts. and Markus, A. S.},
title = {A certain problem about the covering of convex sets with homothetic ones},
journal = {Izv. Mold. Fil. Akad. Nauk SSSR},
volume = {10(76)},
year = {1960},
pages = {87--90},
note = {In Russian},
}
@article{boltyanski1960,
author = {Boltyanski, V. G.},
title = {The problem of illuminating the boundary of a convex body},
journal = {Izv. Mold. Fil. Akad. Nauk SSSR},
volume = {10(76)},
year = {1960},
pages = {77--84},
note = {In Russian},
}
@article{hadwiger1960,
author = {Hadwiger, Hugo},
title = {Ungel{\"o}ste {P}robleme {N}r. 38},
journal = {Elem. Math.},
volume = {15},
year = {1960},
pages = {130--131},
}
@incollection{bezdek2018,
author = {Bezdek, K{\'a}roly and Khan, Muhammad A.},
title = {The geometry of homothetic covering and illumination},
booktitle = {Discrete Geometry and Symmetry},
series = {Springer Proc. Math. Stat.},
volume = {234},
publisher = {Springer},
address = {Cham},
year = {2018},
pages = {1--30},
doi = {10.1007/978-3-319-78434-2_1},
eprint = {1602.06040},
archivePrefix = {arXiv},
}
@article{lassak1984,
author = {Lassak, Marek},
title = {Solution of {H}adwiger's covering problem for centrally symmetric convex bodies in {$E^3$}},
journal = {J. London Math. Soc. (2)},
volume = {30},
number = {3},
year = {1984},
pages = {501--511},
doi = {10.1112/jlms/s2-30.3.501},
}
@article{prymak2023,
author = {Prymak, Andriy},
title = {A new bound for {H}adwiger's covering problem in {$\mathbb{E}^3$}},
journal = {SIAM J. Discrete Math.},
volume = {37},
number = {1},
year = {2023},
pages = {17--24},
doi = {10.1137/22M1490314},
eprint = {2112.10698},
archivePrefix = {arXiv},
}
@article{campos2024,
author = {Campos, Marcelo and van Hintum, Peter and Morris, Robert and Tiba, Marius},
title = {Towards {H}adwiger's conjecture via {B}ourgain slicing},
journal = {Int. Math. Res. Not. IMRN},
volume = {2024},
number = {10},
year = {2024},
pages = {8282--8295},
doi = {10.1093/imrn/rnad198},
eprint = {2206.11227},
archivePrefix = {arXiv},
}
@article{klartag2025,
author = {Klartag, Boaz and Lehec, Joseph},
title = {Affirmative resolution of {B}ourgain's slicing problem using {G}uan's bound},
journal = {Geom. Funct. Anal.},
volume = {35},
number = {4},
year = {2025},
pages = {1147--1168},
doi = {10.1007/s00039-025-00718-w},
eprint = {2412.15044},
archivePrefix = {arXiv},
}
@misc{aim-hadwiger-boltyanski-illumination,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The sharp number of directions illuminating a convex body}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/486-hadwiger-boltyanski-illumination.md},
note = {Entry 486 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-00527, "The sharp number of directions illuminating a convex body", clopen.solutions.