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 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) log² n log log n), where s is the maximum number of intersections between the boundaries of any pair of the given objects.