A characterization theorem for compact unions of two starshaped sets in \({\mathbb R}^ 3\) (Q1073347)

A set S in \({\mathbb{R}}^ d\) has property \(P_ k\) if and only if S is a finite union of d-polytopes and for every finite set F in \(bdry S\) there exist points \(c_ 1,...,c_ k\) (depending on F) such that each point of F is clearly visible via S from at least one \(c_ 1\), \(1\leq i\leq k\). The following results are established. 1) Let \(S\subseteq {\mathbb{R}}^ 3\). If S satisfies property \(P_ 2\), then S is a union of two starshaped sets. 2) Let \(S\subseteq {\mathbb{R}}^ d\), \(d\geq 3\). If S is a compact union of k starshaped sets, then there exists a sequence \(\{S_ j\}\) converging to S (relative to the Hausdorff metric) such that each set \(S_ j\) satisfies property \(P_ k.\) When \(d=3\) and \(k=2\), the converse of 2) above holds as well, yielding a characterization theorem for compact unions of two starshaped sets in \({\mathbb{R}}^ 3\).