Helge Tverberg

Ladies and gentlemen, this is an excellent time to tell you about the beautiful theorem of Tverberg and the startling proof of Sarkaria to Tverberg’s theorem (two parts). A good place to start is Radon’s theorem.

## 1. The theorems of Radon, Helly, and Caratheodory.

### Radon’s theorem

**Radon’s theorem: **Let be points in , . Then there is a partition of such that . (Such a partition is called a **Radon partition**.)

**Proof: **Since the points are affinely dependent. Namely, there are coefficients, not all zero, such that and . Now we can write and , and note that

(*) ,

and also . This last sum is positive because not all the s are equal to zero. We call it .

To exhibit a convex combination of which is equal to a convex combination in just divide relation (*) by . **Walla.**

This trick of basing a partition on the signs of the coefficients repeats in other proofs. Take note!

Radon used his theorem to prove Helly’s theorem.

### Helly’s theorem

**Helly’s theorem**: For a family , , of convex sets in , if every of the sets have a point in common then all of the sets have a point in common.

**Proof:** It is enough to show that when if every of the sets have a point in common then there is a point in common to them all. Let . In words, is a point that belongs to all the s except perhaps to . We assumed that such a point exists for every .

Now we can apply Radon’s theorem: Consider the Radon partition of the points, namely a partition of such that . Let . Since belongs to the convex hull of and every for belongs to every for every not in we obtain that belongs to every for not in . By the same argument belongs to every for not in . Since and are disjoint, the point belongs to all s. **Ahla.**

The proof of Helly’s theorem from Radon’s theorem as described on the cover of a book by Steven R. Lay

### Caratheodory’s theorem

**Caratheodory’s theorem: **For , if then for some , .

Like for Radon’s theorem, there is a simple linear algebra proof. We can assume that is finite; we start with a presentation of as a convex combination of vectors in , and if we can use an affine dependency among the vectors in to change the convex combination, forcing one coefficient to vanish.

Without further ado here is Tverberg’s theorem.

## 2. Tverberg’s Theorem

**Tverberg Theorem (1965):**Let be points in , . Then there is a partition of such that .

So Tverberg’s theorem is very similar to Radon’s theorem except that we want more than two parts! Continue reading →