**Imre Barany, Rade Zivaljevic, Helge Tverberg, and Sinisa Vrecica**** **

Recall the beautiful theorem of Tverberg: (We devoted two posts (I, II) to its background and proof.)

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

The (much easier) case of Tverberg’s theorem is **Radon’s theorem**.

## 1. Eckhoff’s Partition Conjecture

Eckhoff raised the possibility of finding a purely combinatorial proof of Tverberg’s theorem based on Radon’s theorem. He considered replacing the operation : “taking the convex hull of a set ” by an arbitrary closure operation.

Let be a set endowed with an abstract closure operation . The only requirements of the closure operation are:

(1) and

(2) implies .

Define to be the largest size of a (multi)set in which cannot be partitioned into parts whose closures have a point in common.

Eckhoff’s Partition Conjecture: For every closure operation

If is the set of subsets of and is the convex hull operation then Radon’s theorem asserts that and Eckhoff’s partition conjecture would imply Tverberg’s theorem. Update (December 2010): Eckhoff’s partition conjecture was refuted by Boris Bukh. Here is the paper.

## 2. The dimension of Tverberg’s points

For a set , denote by those points in which belong to the convex hull of pairwise disjoint subsets of . We call these points **Tverberg points of order** .

Conjecture (Kalai, 1974):For every , .