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We show that the number of vertices, edges, and faces of the union of k convex polyhedra in 3-space, having a total of n faces, is O(k 3 + kn log k). This bound is almost tight in the worst case, as there exist collections of polyhedra with Ω(k 3 + knα(k)) union complexity. We also describe a rather simple randomized incremental algorithm for computing the… (More)
The paper bounds the combinatorial complexity of the Voronoi diagram of a set of points under certain polyhedral distance functions. Speciically, if S is a set of n points in general position in IR d , the maximum complexity of its Voronoi diagram under the L 1 metric, and also under a simplicial distance function, are both shown to be (n dd=2e). The upper… (More)
We present a simple but powerful new probabilistic technique for analyzing the combinatorial complexity of various substructures in arrangements of piecewi-linear surfaces in higher dimensions. We apply the technique (a) to derive new and simpler proofs of the known bounds on the complexity of the lower envelope, of a single cell, or of a zone in… (More)
The combinatorial complexity of the Voronoi diagram of n lines in three dimensions under a convex distance function induced by a polytope with a constant number of edges is shown to be O(n 2 (n) log n), where is a slowly growing inverse of the Ackermann function. There are arrangements of n lines where this complexity can be as large as (n 2 (n)).