On Grünbaum's conjecture about inner illumination of convex bodies (Q1807613)

One says that a set \(F\) illuminates a convex body \(K\) from within if \(F\) consists of boundary points of \(K\) and for every boundary point \(x\) there is a point \(y\) in \(F\) distinct from \(x\) such that the open line segment \(]x,y[\) is contained in the interior of \(K\). A set \(F\) illuminating a convex body \(K\) from within is called primitive provided that no proper subset of \(F\) illuminates \(K\) from within. In the paper under review it is shown that any primitive illuminating set is finite. The main theorem asserts that for any integers \(m\geq 2\) and \(d\geq 4\) there exists a convex body \(K \subset E^d\) with a primitive illuminating set consisting of at least \(m\) points. This result provides a negative answer to a conjecture of Grünbaum and is in sharp contrast with the cases \(d\leq 3\), where any primitive illuminating set of a convex body in \(E^d\) has at most \(2^d\) points [see \textit{V. Soltan}, Acta Math. Hungar. 69, 15-25 (1995; Zbl 0861.52001)].