Helly's theorem

Helly's theorem is a result about convex shapes. It says that if you have several convex sets (like polygons or disks) in the plane, and every group of three of them has a point in common, then all of them have at least one point in common.

Important properties

• Applies only to convex sets.

• In the plane, checking intersections for every three sets is enough to guarantee a common point for all.

• The theorem can be generalized to higher dimensions: in $n$ dimensions, you check every $n+1$ sets.