An optimal generalization of the centerpoint theorem, and its extensions

We prove an optimal generalization of the centerpoint theorem: given a set P of n points in the plane, there exist two points (not necessarily among input points) that hit all convex objects containing more than 4n/7 points of P. We further prove that this bound is tight. We get this bound as part of a more general procedure for finding small number of points hitting convex sets over P, yielding several improvements over previous results. Categories and Subject Descriptors G.2 [Combinatorics]: Discrete Geometry General Terms Algorithms, Theory Keywords Combinatorial geometry, weak -nets, centerpoint theorem, discrete geometry, extremal methods, hitting convex sets

• Type: Conference
• Year: 2007
• Where: COMPGEOM
• Authors: Saurabh Ray, Nabil H. Mustafa
• Keywords: Centerpoint Theorem, COMPGEOM 2007, Convex Sets, Discrete Geometry, Points Hitting Convex