On Grünbaum's problem about inner illumination of convex bodies (Q1912646)

By an illumination set of a \(d\)-dimensional convex body \(K\) there is understood a subset \(F \subset bdK\) such that for every \(x\in bdK\) there exists a \(y\neq x\) in \(F\) such that the open interval \(]x,y[\) is contained in \(\text{int} K\). \(F\) is called primitive if it does not contain properly another illumination set. Grünbaum conjectured that a primitive illumination set contains at most \(2^d\) points. The author proves the conjecture for \(d=3\), equality holding only for \(K\) being combinatorially equivalent to a cube.