Keiko Imai, Akitoshi Kawamura, Jiří Matoušek, Yu Muramatsu and Takeshi Tokuyama. Distance k-sectors and zone diagrams. 25th European Workshop on Computational Geometry (EuroCG), Brussels, Belgium, March 2009, pages 191–194 of the (unpublished) proceedings.
Fulltext.pdf (A4 paper, two columns, 4 pages) and Errata.pdf
Slide.pdf (presented on March 17, 2009; Q's and A's added on March 28, 2009)
We prove the existence and uniqueness of the zone diagram of a given set of positively separated sites in Euclidean space. This was known for point sites in the plane, but our proof is simpler even for this specific case. We also show the existence of a distance k-sector between two sites. Both proofs rely on the Knaster–Tarski theorem on fixed points of monotone functions.