Monday, October 31st

Problem


A strip of width w is the set of all points which lie on, or between, two parallel lines distance w apart. Let $S$ be a set of $n \geq 3$ points on the plane such that any three different points of $S$ can be covered by a strip of width 1. Prove that S can be covered by a strip of width 2. 


Solution  

Since we have a finite set, there is a pair of points in $S$ which are the furthest apart from each other. Then every point in $S$ is in a strip  of width 1 with one of its sides being the line passing through the maximal pair. Hence $S$ lies inside the union of two such possible strips.