Abstract. 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, and (b) (γ,α) -substantially fat trihedral wedges is at most O(n^ 2+ ɛ ) , for any ɛ >0 , where the constants of proportionality depend on ɛ , 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.
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Pach, ., Safruti, . & Sharir, . The Union of Congruent Cubes in Three Dimensions . Discrete Comput Geom 30, 133–160 (2003). https://doi.org/10.1007/s00454-003-2928-3
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DOI: https://doi.org/10.1007/s00454-003-2928-3