Crossing Families

Given a set of points in the plane, a crossing family is a collection of line segments, each joining two of the points, such that any two line segments intersect internally. Two sets A and B of points in the plane are mutually avoiding if no line subtended by a pair of points in A intersects the convex hull of B, and vice versa. We show that any set of n points in general position contains a pair of mutually avoiding subsets each of size at least n/12. As a consequence we show that such a set possesses a crossing family of size at least n/12, and describe a fast algorithm for finding such a family. AMS classification: 52C10; also 68Q20

• Type: Conference
• Year: 1991
• Where: COMPGEOM
• Authors: Boris Aronov, Paul Erdös, Wayne Goddard, Daniel J. Kleitman, Michael Klugerman, János Pach, Leonard J. Schulman
• Keywords: COMPGEOM 1991, Convex Hull, Discrete Geometry, Line Segments, Vice Versa