We arrived to one the main topics of my thesis: Helly’s theorem. For the sake of simplicity I will explain it in 2-dimension. I will try to use very little of the usual formal math language.

**Helly’s theorem:** We have some convex shapes drawn in the plane and we know that any three of them have a point in common. Then all of them have a point in common.

Ok, but what are **convex shapes**? Well, here are some:

And these are **not** convex:

So, a shape is convex if it has no “dents” in it. More precisely a shape is convex if it contains the segment between any two points of the shape.

Helly’s theorem is a classical result in discrete geometry, and many variations and generalizations are known. In my thesis I create a new generalization.

**Try it for yourself! Try to draw four convex shapes such that any three of them have a point in common, but not all of them. What happens if we use arbitrary shapes, instead of convex ones?**

(If someone is really interested in my thesis, you can read it here: https://www.cs.elte.hu/blobs/diplomamunkak/msc_mat/2017/damasdi_gabor.pdf)