A Krasnosel'skii theorem for nonclosed sets in \({\mathbb{R}}^ d\) (Q1073346)

The following theorem is proved. For each \(d\geq 2\), define \(f(d)=d^ 2- 2d+3\) if \(d\neq 3\) and \(f(3)=7\). Let S be a nonempty bounded set in \({\mathbb{R}}^ d\) and assume that \((cl S)\setminus S\) is a finite union of convex components, each having closure a polytope. If every f(d) points of S see via S a common point, then there is a point p in \(cl S\) such that \(B_ p\equiv \{s: s\quad in\quad S\quad and\quad (p,s)\not\subseteq S\}\) is nowhere dense in S. (By definition, x sees y via S if the segment [x,y] lies in S.) This theorem extends earlier results of the author [J. Geom. 18, 28-42 (1982; Zbl 0503.52007); ibid. 21, 97-100 (1983; Zbl 0525.52010)] from the plane to higher dimensions.