Sylvester-Gallai-like Theorems for Polygons in the Plane

Given an arrangement of lines in the plane, an ordinary point is a point of intersection of precisely two of the lines. Motivated by a desire to understand the fine structure of ordinary points in line arrangements, we consider the following problem: given a polygon P in the plane and a family of lines passing into the interior of P, how many ordinary intersection points must there be on or inside of P? We answer this question for a variety of different types of polygons. In a similar spirit, we investigate bichromatic arrangements of lines intersecting the interior of polygons and determine when such arrangements must have monochromatic intersection points.

By: Jonathan Lenchner

Published in: RC25263 in 2012


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