On the union of fat tetrahedra in three dimensions

We show that the combinatorial complexity of the union of <i>n</i> “fat” tetrahedra in 3-space (i.e., tetrahedra all of whose solid angles are at least some fixed constant) of arbitrary sizes, is <i>O</i>(<i>n</i>^2+ϵ), for any ϵ > 0;the bound is almost tight in the worst case, thus almost settling a conjecture of Pach et al. [2003]. Our result extends, in a significant way, the result of Pach et al. [2003] for the restricted case of <i>nearly congruent cubes</i>. The analysis uses cuttings, combined with 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 <i>dihedral</i> wedges in Δ. As an immediate corollary, we obtain that the combinatorial complexity of the union of <i>n</i> cubes in R³, having arbitrary side lengths, is <i>O</i>(<i>n</i>^2+ϵ), for any ϵ > 0 (again, significantly extending the result of Pach et al. [2003]). Finally, 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 R³.