Math Conjectures

Hadwiger–Levi Illumination Conjecture

OPENLandmarkConjectureProposed 1957–1960 · Standard version

Canonical statement

Let d≥2d\geq2, let Sd−1S^{d-1} be the unit sphere, and let K⊂RdK\subset\mathbb R^d be a convex body. Say that a direction u∈Sd−1u\in S^{d-1} illuminates x∈∂Kx\in\partial K if x+tu∈int⁡Kx+tu\in\operatorname{int}K for some t>0t>0. There exist at most 2d2^d directions such that every point of ∂K\partial K is illuminated by at least one of them.
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Let \(d\geq2\), let \(S^{d-1}\) be the unit sphere, and let
\(K\subset\mathbb R^d\) be a convex body. Say that a direction
\(u\in S^{d-1}\) illuminates \(x\in\partial K\) if
\(x+tu\in\operatorname{int}K\) for some \(t>0\). There exist at most
\(2^d\) directions such that every point of \(\partial K\) is illuminated
by at least one of them.
DG-006:[Hadwiger1957Illumination][Bezdek2010Illumination][RotemSchejterSlomka2026Illumination]
Discrete geometryGeometry and topology

Notes

The illumination conjecture of Hadwiger and Levi concerns lighting the boundary of a convex body K⊂RdK\subset\mathbb R^d from outside: a direction uu illuminates a boundary point xx if the ray from xx in direction uu immediately enters the interior of KK. The conjecture asserts that 2d2^d directions always suffice to illuminate all of ∂K\partial K. It was posed between 1957 and 1960 in equivalent illumination and covering formulations, the covering version asking for 2d2^d smaller homothetic copies of KK whose union covers KK [Hadwiger1957Illumination].

The bound 2d2^d would be sharp: for a parallelotope, each of its 2d2^d vertices needs a direction of its own. The conjecture is known in low dimensions and for many special classes of bodies, and the partial results are surveyed by Bezdek [Bezdek2010Illumination]. Recent work has also transplanted the question to new settings, such as the complex illumination problem [RotemSchejterSlomka2026Illumination].

For arbitrary convex bodies in dimensions d≥3d\ge3 no universal proof is known, and the conjecture remains open.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.