The Limit of Lp Voronoi Diagrams as p→0 is the Bounding-Box-Area Voronoi Diagram

• Published in: Discrete & computational geometry Vol. 72; no. 3; pp. 1284 - 1303
• Main Authors: Haverkort, Herman, Klein, Rolf
• Format: Journal Article
• Language: English
• Published: New York Springer Nature B.V 01.10.2024
• Subjects: Hyperbolas; Voronoi graphs
• ISSN: 0179-5376; 1432-0444
• DOI: 10.1007/s00454-023-00599-6

We consider the Voronoi diagram of points in the real plane when the distance between two points a and b is given by Lp(a-b) where Lp((x,y))=(|x|p+|y|p)1/p. We prove that the Voronoi diagram has a limit as p converges to zero from above or from below: it is the diagram that corresponds to the distance function L∗((x,y))=|xy|. In this diagram, the bisector of two points in general position consists of a line and two branches of a hyperbola that split the plane into three faces per point. We propose to name L∗ as defined above the geometric L0 distance.