**Tverberg Theorem (1965):** Let $ x_1,x_2,\dots, x_m$ be points in $ R^d$, $ m \ge (r-1)(d+1)+1$. Then there is a partition $ S_1,S_2,\dots, S_r$ of $ \{1,2,\dots,m\}$ such that $\cap _{j=1}^rconv (x_i: i \in S_j) \ne \emptyset$.

Tverberg's theorem was conjectured by Birch who also proved the planar case. The case $r=2$ is a 1920 theorem of Radon which follows easily from linear algebra consideration.

(The first thing to note is that Tverberg's theorem is sharp. If you have only $ (r-1)(d+1)$ points in $ R^d$ in a "generic" position then for every partition into $ r$ parts even the affine spans of the points in the parts will not have a point in common.)

The first proof of this theorem appeared in 1965. It was rather complicated and was based on the idea to first prove the theorem for points in some special position and then show that when you continuously change the location of the points the theorem remains true. A common dream was to find an extension of the proof of Radon's theorem, a proof which is based on the two types of numbers - positive and negative. Somehow we need three, four, or $ r$ types of numbers. In 1981 Helge Tverberg found yet another proof of his theorem. This proof was inspired by Barany's proof of the colored Caratheodory theorem (mentioned below) and it was still rather complicated. It once took me 6-7 hours in class to present it.

What could be the probability of hearing two new simple proofs of Tverberg'stheorem on the same day? While visiting the Mittag-Leffler Institute in 1992, I met Helge one day around lunch and asked him if he has found a new proof. To my surprise, he told me about a new proof that he found with Sinisa Vrecica. This is a proof that can be presented in class in 2 hours! It appeared (look here) along with a far-reaching conjecture (still unproved). Later in the afternoon I met Karanbir Sarkaria and he told me about a proof he found to Tverberg's theorem which was absolutely startling. This is a proof you can present in a one hour lecture; it also somehow goes along with the dream of having $r$ "types" of numbers replacing the role of positive and negative real numbers. Another very simple proof of Tverberg's theorem was found by Jean-Pierre Roudneff in 1999.

For further details see these blog posts (I,II).

didn'tthink that the proof could be much improved, but then were proven wrong. $\endgroup$ – David Corwin Jul 9 '12 at 6:02