Visible shorelines containing at least \(k\) vertices (Q1587841)

A point \(p\) in a compact set \(C\subset\mathbb{R}^d\) is visible from \(x\in\mathbb{R}^d\setminus C\) via the complement of \(C\) if \(p\) is the only intersection of the segment \([x,p]\) with \(C\). For any convex polygon in \(\mathbb{R}^2\) and any set \(S\subset \mathbb{R}^2 \setminus C\) it is proved that if every three points of \(S\) view at least \(k\) common vertices of \(C\) via the complement of \(C\), \(k\geq 3\), then there are \(k\) vertices of \(C\) which are visible from all the points of \(S\). Further for \(k\geq 3\) the number three turns out to be the best possible, while in the cases \(k=1,2\) triples of points have to be replaced by at least five, four points, respectively.