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On bisectors for convex distance functions in 3-space

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.