# Sarkaria’s Proof of Tverberg’s Theorem 2

Karanbir Sarkaria

## 4. Sarkaria’s proof:

Tverberg’s theorem (1965): Let $v_1, v_2,\dots, v_m$ 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 $\cap _{j=1}^rconv (v_i: i \in S_j) \ne \emptyset$.

Proof: We can assume that $m=(r-1)(d+1)+1$. First suppose that the points $v_1, v_2,\dots,v_m$ belong to the $d$-dimensional affine space $H$ in $V=R^{d+1}$ of points with the property that the sum of all coordinates is 1. Next consider another $(r-1)$-dimensional vector space $W$ and $r$ vectors in $W$, $w_1,w_2, \dots, w_r$  such that $w_1+w_2+\dots +w_r = 0$  is the only linear relation among the $w_i$s. (So we can take $w_1,\dots w_{r-1}$ as the standard basis in $R^{r-1}$ and $w_r=-w_1-w_2-\dots-w_r$. )

Now we consider the tensor product $V \otimes W$.

Nothing to be scared of: $U=V \otimes W$ can be regarded just as the $(d+1)(r-1)$-dimensional vector space of d+1  by $r-1$ matrices. We can define the tensor product  $x \otimes y$ of two vectors $x = (x_1,x_2,\dots,x_{d+1}) \in V$ and $y =(y_1,y_2,\dots,y_{r-1}) \in W$, as the (rank-one) matrix $(x_i \cdot y_j)_{1 \le i \le d+1,1 \le j \le r-1}$.

Consider in U the following sets of points:

$S_1 = \{v_1 \otimes w_j: j=1,2,\dots r \}$

$S_2 = \{v_2 \otimes w_j: j=1,2,\dots r \}$

$S_i = \{v_i \otimes w_j: j=1,2,\dots r \}$

$S_m = \{v_m \otimes w_j: j=1,2,\dots r \}.$

Note that 0 is in the convex hull of every $S_i$. Indeed it is the sum of the elements in $S_j$. (And thus $0=1/r(v_i \otimes w_1+ v_i \otimes w_2 + \dots + v_i \otimes w_r)$. )

By the Colorful Caratheodory Theorem we can  Continue reading