Illumination and visibility problems in terms of closure operators (Q1878965)

This paper continues the subject considered earlier by these authors in [Aequationes Math. 64, 128--135 (2002; Zbl 1013.52005)]. Recall that a point \(x\) of the boundary \(\partial (K)\) of a convex body \(K \subset R^n\) is called to be illuminated by a point \(z \not \in K\) if there exists a point \(y \in \text{ int} (K)\) such that \(x \in yz\). Moreover, we say that a set \(S\) disjoint with \(K\) illuminates \(\partial (K)\) if every point of \(\partial (K)\) is illuminated by a point of \(S\). The authors present a few conditions equivalent to the second definition. They are expressed in terms of closure operators.