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

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.