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Selecting Points that are Heavily Covered by Pseudo-Circles, Spheres or Rectangles

Published online by Cambridge University Press:  28 April 2004

SHAKHAR SMORODINSKY
Affiliation:
School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: smoro@tau.ac.il)
MICHA SHARIR
Affiliation:
School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel, and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA (e-mail: sharir@cs.tau.ac.il)

Abstract

In this paper we prove several point selection theorems concerning objects ‘spanned’ by a finite set of points. For example, we show that for any set $P$ of $n$ points in $\R^2$ and any set $C$ of $m \,{\geq}\, 4n$ distinct pseudo-circles, each passing through a distinct pair of points of $P$, there is a point in $P$ that is covered by (i.e., lies in the interior of) $\Omega(m^2/n^2)$ pseudo-circles of $C$. Similar problems involving point sets in higher dimensions are also studied.

Most of our bounds are asymptotically tight, and they improve and generalize results of Chazelle, Edelsbrunner, Guibas, Hershberger, Seidel and Sharir [8], where weaker bounds for some of these cases were obtained.

Type
Paper
Copyright
2004 Cambridge University Press

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