# On Mutually Avoiding Sets

• Pavel Valtr
Chapter
Part of the Algorithms and Combinatorics book series (AC, volume 14)

## Summary

Two finite sets of points in the plane are called mutually avoiding if any straight line passing through two points of anyone of these two sets does not intersect the convex hull of the other set. For any integer n, we construct a set of n points in general position in the plane which contains no pair of mutually avoiding sets of size more than $$O\left( {\sqrt n } \right)$$ each. The given bound is tight up to a constant factor, since Aronov et al. [1] showed a polynomial-time algorithm for finding two mutually avoiding sets of size $$\Omega \left( {\sqrt n } \right)$$ each in any set of n points in general position in the plane.

## Keywords

Line Segment Convex Hull General Position Geometric Graph Convex Quadrilateral
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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