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Discrete Comput Geom 4:239-243 (1989) C [h.,'rvte~COntlmtala~4tal Jeometrv~. 1989 Sprmger-Vedag New York In
 

Summary: Discrete Comput Geom 4:239-243 (1989) C [h.,'rvte~COntlmtala~4tal
Jeometrv~. 1989 Sprmger-Vedag New York Iną
Cutting Disjoint Disks by Straight Lines*
N. Alon, ] M. Katchalski, 2 and W. R. Pulleyblank 3
Department of Mathematics, Tel Aviv University, Tel Aviv, Israel
2Department of Mathematics, Technion-lsrael Institute of Technology, Haifa, Israel
3Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada
Abstract. For k > 0 let f(k) denote the minimum integerf such that, for any family
of k pairwise disjoint congruent disks in the plane, there is a direction t~ such that
any line having direction ~ intersects at mostf of the disks. We determine the exact
asymptotic behavior off(k) by proving that there are two positive constants dt, d 2
such that dlx/k.,/~gk~f(k)~d:,x/k~,/~gk. This result has been motivated by
problems dealing with the separation of convex sets by straight lines.
1. Introduction
For k > 0 let f(k) denote the minimum integer f such that, for any family A of
k pairwise disjoint unit radius disks in the plane, there is a direction a such that
any straight line having direction a intersects at most f of the disks. In this note
the following result is proved.
Theorem 1.1. There exist two positive constants dl, d2 such that
dl,J-k ' v/~g k <-f( k ) <-d2x/k . x/~g k

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics