Opposite-Quadrant Depth in the Plane

Authors:
Hervé Brönnimann;Jonathan Lenchner;János Pach
Affiliations:
Polytechnic University, Computer and Information Science, 11201, Brooklyn, NY, USA;IBM T.J. Watson Research Center, 10598, Yorktown Heights, NY, USA;Courant Institute, NYU, and City College, CUNY 251 Mercer Street, 10012, New York, NY, USA
Venue:
Graphs and Combinatorics
Year:
2007

Citing 0
Cited 3

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Abstract

Given a set S of n points in the plane, the opposite-quadrant depth of a point p∈S is defined as the largest number k such that there are two opposite axis-aligned closed quadrants (NW and SE, or SW and NE) with apex p, each quadrant containing at least k elements of S. We prove that S has a point with opposite-quadrant depth at least n/8. If the elements of S are in convex position, then we can guarantee the existence of an element whose opposite-quadrant depth is at least n/4. Both results are asymptotically best possible.