On the union of fat tetrahedra in three dimensions
Journal of the ACM (JACM)
Tangencies between families of disjoint regions in the plane
Proceedings of the twenty-sixth annual symposium on Computational geometry
Tangencies between families of disjoint regions in the plane
Computational Geometry: Theory and Applications
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A compact set c in ${\Bbb R}^d$ is κ-round if for every point $p\in \partial c$ 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 ${\Bbb R}^3$ (resp., in ${\Bbb R}^4$) of constant description complexity is O(n2+ε) (resp., O(n3+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.