Convex distance functions in 3-space are different

We investigate the bisector systems of convex distance functions in 3-space and show that there is a substantial difference to the Euclidean metric which cannot be observed in 2-space. Namely, more than one sphere can pass through four points in general position. We show that in the L4-metric there exist quadrupels of points that lie on the surface of three L4-spheres, and that this number does not decrease if the four points are disturbed independently within 3-dimensional neighborhoods. Moreover, for each n ≥ 2 we construct a smooth and symmetric convex distance function d and four points that are contained in the surface of exactly n d-spheres. This result implies that there is no general upper bound to the complexity of the Voronoi diagram of four sites based on a convex distance function in 3-space.

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