Im studying the book "Lectures on Discrete Geometry" of Jiri Matousek.
The chapter 8.3 is about the Theorem of Tverberg which says: Let d and r natural numbers. For any set $A\subset R^d$ of at least (d+1)(r-1)+1 points there exist r pairwise disjoint subsets $A_1,...,A_r$$\subset A$ such that the intersection of all the convex hulls of $A_i$ is different of the empty set.
In the exercise problems there is the next one that I tried a lot of things but I can't solve and it's frustrating because I already solved the other exercises which are supposed to be more difficult than this:
- Prove (directly, withouth using Tverberg's theorem) that for any integers $d,r_1,r_2$ we have $T(d,r_1r_2)$$<=T(d,r_1)T(d,r_2)$
Where $T(d,r)$ is the number of points that you need in dimension $d$ to get partitioned in $r$ pairwise disjoint sets such that the intersection of all the convex hulls is different of the empty set.
Sorry for my bad Latex and probably english.