Small Strong Epsilon Nets

Let P be a set of n points in R-d. A point x is said to be a centerpoint of P if x is contained in every convex object that contains more than dn/d+1 points of P. We call a point x a strong centerpoint for a family of objects C if x is an element of P is contained in every object C is an element of C that contains more than a constant fraction of points of P. A strong centerpoint does not exist even for halfspaces in R-2. We prove that a strong centerpoint exists for axis-parallel boxes in Rd and give exact bounds. We then extend this to small strong epsilon-nets in the plane. Let epsilon(S)(i) represent the smallest real number in [0, 1] such that there exists an epsilon(S)(i)-net of size i with respect to S. We prove upper and lower bounds for epsilon(S)(i) where S is the family of axis-parallel rectangles, halfspaces and disks. (C) 2014 Elsevier B.V. All rights reserved.