On voronoi diagrams in the Lp-metric in higher dimensions

Abstract

We prove upper bounds on the number of Lp-spheres passing through D+1 points in general position in D-space, and on the sum of the Betti numbers of the intersection of bisectors in the Lp-metric, where p is an even positive integer. The bounds found, surprisingly, do not depend on p. The proofs for these bounds involve the techniques of Milnor [14] and Thorn [20] for finding bounds 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, 16]. Furthermore, using the theory of degree of mappings in D-space we prove that for even p the number of Lp-spheres passing through D+1 points in general position is odd. Combined with results in [10, 11], our results clarify the structure of Voronoi diagrams based on the Lp-metric (with even p) in 3-space.