Levi-Hadwiger illumination conjecture

Let $n\ge 2$ and let $K\subseteq\mathbb{R}^n$ be a convex body. The illumination number $\operatorname{ill}(K)$ is the minimum number of directions needed to illuminate $K$.

Levi-Hadwiger illumination conjecture. We have

$$\operatorname{ill}(K)\le 2^n.$$

Moreover, equality holds if and only if $K$ is an affine image of the hypercube.

By Boltyanski's equivalence, this is equivalent to the Levi-Hadwiger covering problem, which asks whether every convex body can be covered by at most $2^n$ translates of its interior, with equality only for affine images of the hypercube. The conjecture is unresolved in general.