The union of congruent cubes in three dimensions

A dihedral (trihedral) wedge is the intersection of two (resp. three) half-spaces in R³. It is called α-fat if the angle (resp., solid angle) determined by these half-spaces is at least α > 0. If, in addition, the sum of the three face angles of a trihedral wedge is at least γ > 4π/3, then it is called (γ, α)-substantially fat. We prove that, for any fixed γ > 4π/3, α > 0, the combinatorial complexity of the union of n (a) α-fat dihedral wedges, (b) (γ, α)-substantially fat trihedral wedges is at most O(n^2+ε), for any ε > 0, where the constants of proportionality depend on ε, α (and γ). We obtain as a corollary that the same upper bound holds for the combinatorial complexity of the union of n (nearly) congruent cubes in R³. These bounds are not far from being optimal.