# Seven Problems Around Tverberg’s Theorem

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 $x_1,x_2,\dots, x_m$ 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 $\cap _{j=1}^rconv (x_i: i \in S_j) \ne \emptyset$.

The (much easier) case $r=2$ 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 $A$” by an arbitrary closure operation.

Let $X$  be a set endowed with an abstract closure operation $X \to cl(X)$. The only requirements of the closure operation are:

(1) $cl(cl (X))=cl(X)$ and

(2) $A \subset B$ implies $cl(A) \subset cl (B)$.

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

Eckhoff’s Partition Conjecture: For every closure operation $t_r \le t_2 \cdot (r-1).$

If $X$ is the set of subsets of $R^d$ and $cl(A)$ is the convex hull operation then Radon’s theorem asserts that $t_2(X)=d+1$ 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 $A$, denote by $T_r(A)$ those points in $R^d$ which belong to the convex hull of $r$ pairwise disjoint subsets of $X$. We call these points Tverberg points of order $r$.

Conjecture (Kalai, 1974): For every $A \subset R^d$ , $\sum_{r=1}^{|A|} {\rm dim} T_r(A) \ge 0$.