Almost tight bound for the union of fat tetrahedra in three dimensions

We show that the combinatorial complexity of the union of n "fat" tetrahedra in 3-space (i.e., tetrahedra all of whose solid angles are at least some fixed constant) of arbitrary sizes, is O(n_2+ε), for any ε > 0; the bound is almost tight in the worst case, thus almost settling a conjecture of Pach et al. [24]. Our result extends, in a significant way, the result of Pach et al. [24] for the restricted case of nearly congruent cubes. The analysis uses cuttings, combinedwith the Dobkin-Kirkpatrick hierarchical decomposition of convex polytopes, in order to partition space into subcells, so that, on average, the overwhelming majority of the tetrahedra intersecting a subcell Δ behave as fat dihedral wedges in Δ. As an immediate corollary, we obtain that the combinatorial complexity of the union of n cubes in IR³, having arbitrary side lengths, is O(n^2+ε), for any ε ≥ 0 (again, significantly extending the result of [24]). Our analysis can easily be extended to yield a nearly-quadratic bound on the complexity of the union of arbitrarily oriented fat triangular prisms (whose cross-sections have arbitrary sizes) in IR³. Finally, we show that a simple variant of our analysis implies a nearly-linear bound on the complexity of the union of fat triangles in the plane.