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A tight bound for the complexity of Voronoi diagrams under polyhedral convex distance functions in 3D
 PROC. 33RD ANNU. ACM SYMPOS. THEORY OF COMPUT
, 2001
"... We consider the Voronoi diagram of a set of n points in three dimensions under a convex distance function induced by an arbitrary, fixed polytope. The combinatorial complexity, i. e. the number of vertices, edges, and facets, of this diagram is shown to be in Θ(n²), which constitutes a considerable ..."
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We consider the Voronoi diagram of a set of n points in three dimensions under a convex distance function induced by an arbitrary, fixed polytope. The combinatorial complexity, i. e. the number of vertices, edges, and facets, of this diagram is shown to be in Θ(n²), which constitutes a considerable improvement to the results known so far. Unlike previous work, we do not need probabilistic arguments or recurrence techniques, but we exploit properties of the involved geometric structures. Key observations are that the lower envelope of n polygonal chains in the plane with a total of O(n) line segments and with only a fixed number of slopes is linear in n, and that the number of slopes of Voronoi edges is bounded by a constant that only depends on the complexity of the polytope.
On bisectors for convex distance functions in 3space
 In Proc. 11th Canad. Conf. Comput. Geom
, 1999
"... We investigate the structure of the bisector of point sites under arbitrary convex distance functions in three dimensions. Our results show that it is advantageous for analyzing bisectors to consider their central projection on the unit sphere, thereby reducing by one the dimension of the problem. F ..."
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Cited by 2 (1 self)
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We investigate the structure of the bisector of point sites under arbitrary convex distance functions in three dimensions. Our results show that it is advantageous for analyzing bisectors to consider their central projection on the unit sphere, thereby reducing by one the dimension of the problem. From the concept of “silhouettes ” and their intersections we obtain simple characterizations of important structural properties like the number of connected components of the bisector of three sites. Furthermore, we prove that two related bisectors of three sites may intersect in permuted order.