Illumination by translates of convex sets (Q1189648)

For \(S\subset \mathbb{R}^ d\) and \(x,y\in S\) we say that \(x\) is visible from \(y\) via \(S\) if \([x,y]\subset S\). Similarly, \(x\) is clearly visible from \(y\) via \(S\) if there is some neighbourhood \(N\) of \(x\) such that each point of \(N\cap S\) is visible from \(y\) via \(S\). For \(T\subset\mathbb{R}^ d\) and \(S'\subset S\), the set \(T\) is said to illumine \(S'\) via \(S\). (One can say that \(T\) is an illuminator for \(S'\) via \(S\).) Likewise, \(T\) clearly illumines \(S'\) via \(S\) if each point of \(S'\) is clearly visible via \(S\) from some point of \(T\). A famous theorem of Krasnosel'skii (1946) states that the non-empty compact set \(S\subset \mathbb{R}^ d\) is star-shaped iff every \(d+1\) points of \(S\) are visible via \(S\) from a common point (for references and variations see the interesting discussion of \textit{H. T. Croft}, \textit{K. J. Falconer}, and \textit{R. K. Guy} in section E 2 of their recent book ``Unsolved Problems in Geometry'', Springer, New York Inc. (1991)). In the present paper, analogues of this theorem are investigated. (Instead of showing that \(S\) is star-shaped, the aim is to show that \(S\) has a convex illuminator). E.g., if some translate of the compact convex set \(T\subset R^ 2\) clearly illuminates every three points of \(S\subset R^ 2\), then there is a translate of \(T\) which illumines each point of the compact set \(S\). The paper contains interesting analogues in higher dimensions, too. It should be underlined that such results have applications in geometric location theory, recently also from the viewpoint of computational geometry.