Local characterization of starshapedness
Let \(S\) be a subset of \(E^d\). The following definitions hold; the point \(x\) sees \(y\) via \(S\) if the closed segment \([x,y] \subset S\); \(st(x,S)\) denotes the set of all points of \(S\) that see \(x\) via \(S\); \(x\) sees clearly \(y\) via \(S\) if \(y\) admits a neighbourhood \({\mathcal N}\) such that \(x\) sees the whole of \({\mathcal N} \cap S\) via \(S\); \(x\) has higher visibility via \(S\) than \(y\) if \(st(y,S) \subseteq st(x,S)\); \(x\) is a point of maximal visibility if \(x\) admits a neighbourhood \({\mathcal N}\) such that \(x\) has higher visibility via \(S\) than any other point of \({\mathcal N} \cap S\); a hunk is a set \(S\) such that \(S\) is connected and \(S=\text{cl} \int S\); a star center of \(S\) is a point \(x\in S\) such that \(st(x,S) =S\); \(\ker S\) is the set of all star-centers of \(S\). Hence \(S\) is starshaped iff \(\ker S\) is not empty. The authors prove that if \(S\) is a compact hunk in \(E^d\) and \(x\in S\), \(x\) is a point of maximal visibility iff \(x\in\ker S\). From this result follows that if \(S\) is a compact hunk and has a point of maximal visibility, \(S\) is starshaped.
scientific article; zbMATH DE number 1044129