This week, we learn about Helly's Theorem, a surprising result from combinatorial geometry which explains how convex sets may intersect each other.
How would you use Helly's Theorem to solve the following?
Let be a finite set of points in the plane, such that the distance between any 2 points is less than 1. Show that there exists a disk of radius which covers all of these points.
Show that any polygon with perimeter 2 can be covered with a disk of radius 1.
What if the edges do not need to be straight lines?