A planar 3-convex set is indeed a union of six convex sets
For a planar set \(S\), two points \(a,b \in S\) see each other via \(S\) if \([a,b]\) is included in \(S\). A planar set \(S\) is called \(3\)-convex, if for every three points of \(S\), at least two see each other via \(S\). By a result of \textit{F.~A.~Valentine} [Pac. J. Math. 7, 1227--1235 (1957; Zbl 0080.15401)], every closed \(3\)-convex \(S\) is a union of three convex sets. The authors of the paper under review drop the condition that \(S\) is closed and show that \(S\) is a union of (at most) six convex sets (the number six is best possible). It should be noted that the authors complete a result of \textit{M.~Breen} [Pac. J. Math. 53, 43--57 (1974; Zbl 0287.52003)], who claimed this property earlier.
scientific article; zbMATH DE number 6171522