A characterization theorem for certain unions of two starshaped sets in \({\mathbb{R}}^ 2\) (Q1087141)

For x and y in \(S\subset {\mathbb{R}}^ d\), x is called clearly visible from y via S if there is some neighbourhood N of x such that [y,z]\(\subset S\) for each \(z\in N\cap S\). For a compact simple connected (complement has one component) set S in \({\mathbb{R}}^ 2\), it is shown that S is the union of two starshaped sets iff for every finite F in the boundary of S there is a set G also in the boundary of S and arbitrarily close of F and there are points s, t such that each point of G is clearly visible via S from one of s,t.