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Small strong epsilon nets

Ashok, Pradeesha and Azmi, Umair and Govindarajan, Sathish (2014) Small strong epsilon nets. In: COMPUTATIONAL GEOMETRY-THEORY AND APPLICATIONS, 47 (9). pp. 899-909.

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Official URL: http://dx.doi.org/ 10.1016/j.comgeo.2014.05.002

Abstract

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.

Item Type: Journal Article
Additional Information: Copy right for this article belongs to the ELSEVIER SCIENCE BV, PO BOX 211, 1000 AE AMSTERDAM, NETHERLANDS
Keywords: Centerpoint; epsilon-Nets; Axis-parallel rectangles; Small weak epsilon-nets
Department/Centre: Division of Electrical Sciences > Computer Science & Automation
Depositing User: Id for Latest eprints
Date Deposited: 03 Sep 2014 07:21
Last Modified: 03 Sep 2014 07:21
URI: http://eprints.iisc.ac.in/id/eprint/49711

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