Hadwiger’s illumination conjecture

At the top-right corner of a square, moving down and left enters the square. Moving only left stays on the top edge. A direction counts as illuminating a boundary point only when a short move along it reaches the interior. Four diagonal directions suffice for the whole square, and its four corners require different directions. Can every convex solid be illuminated by a similarly limited set: at most four directions in the plane, eight in three dimensions, and a bound that doubles for each extra dimension?[1]

Can a small set of inward directions serve every boundary point of a solid shape? The conjecture asks for at most $2^n$ directions in dimension $n$. The body must be closed and bounded with an interior, and convex: the segment joining any two of its points stays inside it. A direction $v$ illuminates a boundary point $x$ if $x+tv$ lies in the interior for some $t>0$. The direction set may depend on the body; the conjectured count depends only on dimension.

For the square $[-1,1]^2$, its corner $(1,1)$ moves inside along $(-1,-1)$ for sufficiently small positive $t$. Along $(-1,0)$, it stays on the boundary. The four corners require the four strict component-sign combinations, so a single direction cannot serve two different corners. Four diagonal directions also cover the remaining boundary.

An equivalent covering formulation asks for at most $2^n$ translated, positively scaled copies of the body, each smaller than the original. These copies preserve its shape and orientation. The equivalence links choosing inward directions to covering a whole shape with smaller versions of itself.[1]

Some families are understood. The guarantee for every convex body is still missing.[2]

The September 2026 preprint reports a fractional result permitting weighted directions. Its reported integral consequence has extra factors, so it does not prove the target.[2]

Every boundary point imposes an inward-movement requirement, but one direction can serve many points at once. A universal bound would show that even a complicated convex boundary cannot require arbitrarily many separate directions in a fixed dimension. The square's corners show why the count must allow genuinely different local requirements.[1]

Illumination connects local boundary geometry to a global covering question. A sharp dimension-only bound would limit how many smaller copies of a convex shape are needed to cover it, despite variations in curvature, flat faces and corners. Cubes explain why the proposed count cannot be lowered in general: their corners impose incompatible inward sign requirements. The conjecture asks whether any other convex shape can force a larger count.[1]