A closed \((n+1)\)-convex set in \({\mathbb{R}}^ 2\) is a union of \(n^ 6\) convex sets
It is shown that if a closed set S in the plane is \(n+1\)-convex, then it has no more than \(n^ 4\) holes. As a consequence, it can be covered by \(\leq n^ 6\) convex subsets. This is an improvement on the bound of \(2^ n\cdot n^ 3\) obtained by \textit{M. Breen} and \textit{D. C. Kay} [Israel J. Math. 24, 217-233 (1976; Zbl 0342.52006)].
scientific article; zbMATH DE number 4161569