On the union of fat tetrahedra in three dimensions

We showthat the combinatorial complexity of the union of n "efat"e 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. [2003]. Our result extends, in a significant way, the result of Pach et al. [2003] for the restricted case of nearly congruent cubes. 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 behave as fat dihedral wedges in δ. As an immediate corollary, we obtain that the combinatorial complexity of the union of n cubes in ℝ ³, having arbitrary side lengths, is O(n ^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 ℝ ³.