Hadwiger–Boltyanski illumination conjecture for convex bodies

Let $\mathbf{K}$ be a $d$-dimensional convex body in $\mathbb E^d$, where $d\geq 3$, and let $I(\mathbf{K})$ denote the smallest number of directions in $\mathbb S^{d-1}$ that illuminate every boundary point of $\mathbf{K}$. Illumination conjecture. The illumination number satisfies

$$I(\mathbf{K})\leq 2^d,$$

and equality holds only if $\mathbf{K}$ is an affine $d$-cube. This is a longstanding open problem in discrete geometry, known to be solved only in the plane; the conjecture is also related to the equivalent covering and separation conjectures.