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- W206711386 abstract "We prove upper bounds on the number ofL p-spheres passing throughD+1 points in general position in ℝ”, and on the sum of the Betti numbers of the intersection of bisectors in theL p-metric, wherep is an even positive integer. The bounds found do not depend onp. Our result implies that the complexity of Voronoi diagrams (for point sites in general position) in theL p-metric is bounded for increasingp. The proof for this upper bound involves the techniques of Milnor [12] and Thom [16] for finding a bound on the sum of the Betti numbers of algebraic varieties, but instead of the usual degree of polynomials we use their additive complexity, and apply results of Benedetti and Risler [2], [13]. Furthermore, we prove that inD dimensions and for evenp the number ofL p-spheres passing throughD+1 points in general position is odd. In particular, combined with results of [8], [9], our results clarify the structure of Voronoi diagrams based on theL p-metric (with evenp) in three dimensions. For the proof we use the theory of degree of continuous mappings in ℝD, which is a tool widely applied in nonlinear analysis [14]." @default.
- W206711386 created "2016-06-24" @default.
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- W206711386 date "1996-02-01" @default.
- W206711386 modified "2023-10-16" @default.
- W206711386 title "On voronoi diagrams in theL p-metric in ℝD" @default.
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- W206711386 doi "https://doi.org/10.1007/bf02716806" @default.
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