The complexity of the union of (α,β)-covered objects

An (α, β)-covered object is a simply connected planar region c with the property that for each point p ∈ ∂c there exists a triangle contained in c and having p as a vertex, such that all its angles are at least α > 0 and all its edges are at least β.diam(c)-long. This notion extends that of fat convex objects. We show that the combinatorial complexity of the union of n (α, β)-covered objects of "constant description complexity" is O(λ _s+2(n) log ² n log log n), where s is the maximum number of intersections between the boundaries of any pair of given objects, and λ _s (n) denotes the maximum length of an (n, s)-Davenport-Schinzel sequence. Our result extends and improves previous results concerning convex α-fat objects.