Unions of three starshaped sets in \({\mathbb{R}}^ 2\)
Two theorems concerning ``visibility'' are proved. Theorem 1: Let S be a compact simply connected set in \({\mathbb{R}}^ 2\). If every boundary point of S is clearly visible via S from one of the three points a, b, c, then S is a union of three starshaped sets whose kernels contain a, b, c, respectively. The result fails when the number three is replaced by four. Theorem 2: Let S be a compact, simply connected set in \({\mathbb{R}}^ 2\), with points a, b, c in S and point x in bdry S. For some spherical neighborhood N of x, assume that every point of \(N\cap S\) sees via S at least one of a, b or c. Then there exists some \(y\in N\cap bdry S\) clearly visible via S from at least one of a, b or c.
scientific article; zbMATH DE number 4129535