On the union of \(\kappa\)-curved objects (Q1969596)

An object \(C\) in the plane is \(\kappa\)-curved (for \(0 < \kappa < 1\)) if it has constant description complexity, and for each point \(p\) on the boundary of \(C\) one can place a disk \(B\) of radius \(\kappa\operatorname {diam}(C)\) whose boundary passes through \(p\). The authors prove that the combinatorial complexity of the boundary of the union of a set \(\mathcal C\) of \(n\) \(\kappa\)-curved objects is \(O(\lambda_{s}(n)\log n)\), for some constant \(s\), where \(\lambda_{s}\) is the maximum length of an \((n,s)\) Davenport-Schinzel sequence.