Applications of Helly’s theorem

1. Prove that if the plane can be covered with half planes then there exist three of these which also cover the plane.
2. On a circle consider a finite set of arcs which do not cover the circle, such that any two of them have non-void intersection. Show that all arcs have a common points. If the arcs cover the circle does the conclusion still hold?
3. Consider half circles which cover the whole circle. Show that we can pick three of them which still cover the circle.

I’ll not provide the solutions for now. The title should be a strong indication towards finding a solution!

Helly’s Theorem

Helly’s theorem. Let convex figures be given in the plane and suppose each three of them have a common point. Prove that all figures have a common point.

Can the convexity hypothesis be removed? 

Halfplanes cover

Suppose that the plane can be covered by half-planes. Then we can find three of these half-planes which cover the plane.

Helly -> Centring -> Blaschke

Prove that if a finite collection of convex sets in the plane has the property that every three sets from the collection have a point in common, then they all have a point in common. Prove that this holds for an infinite collection also if the sets from the given collection are bounded.

Consider a closed convex figure in plane. Prove that we can find an interior point such that for any chord of the figure passing through we have .

The breadth of a closed convex curve in a given direction is the distance oh the two parallel lines of the given direction which enclose the given curve. The width of a curve the minimum value of the breadth as the direction varies. Prove that a convex curve of width contains a circle of radius , and this is the best value possible.
Reference: J.H Cadwell, Topics in Recreational Mathematics

Rectangles in the plane

We consider a finite family of rectangles in plane, having edges parallel with the coordinate axes such that for any pair of rectangles there exists a vertical line, or a horizontal one which intersects both rectangles. Prove that there exists a vertical line and a horizontal one such that any of the given rectangles intersects at least one of them.