The union of congruent cubes in three dimensions

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