On the union of κ-round objects in three and four dimensions

A compact set c in ℝ^d is κ-round if for every point p ∈ ∂ C there exists a closed ball that contains p, is contained in c, and has radius κ diam c. We show that, for any fixed κ > 0, the combinatorial complexity of the union of n κ-round, not necessarily convex, objects in ℝ³ (resp., in ℝ⁴) of constant description complexity is O(n^2+ε) (resp., O(n ^3+ε)) for any ε > 0, where the constant of proportionality depends on ε, κ, and the algebraic complexity of the objects. The bound is almost tight in the worst case.