The Tverberg Theorem states the following: Let $x_1,x_2,\dots, x_m$ be points in $R^d$ with $m \ge (r-1)(d+1)+1$. 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 bound of $(r-1)(d+1)+1$ in the theorem is sharp because there are point configurations with $(r-1)(d+1)$ points that do not have a Tverberg partition of length $r$.

My question is about lowering this bound by imposing some structure to the points. That is: if we have a full dimensional point configuration $S$ with $m$ points in $R^d$ such that $m\leq(r-1)(d+1)$, can we put conditions on $S$ which still guaratee the existence of a Tverberg partition of length $r$?

Gil Kalai has some very nice posts on the Tverberg Theorem in his blog: http://gilkalai.wordpress.com/2008/11/24/sarkarias-proof-of-tverbergs-theorem-1/ , http://gilkalai.wordpress.com/2008/11/26/sarkarias-proof-of-tverbergs-theorem-2/ and http://gilkalai.wordpress.com/2008/12/23/seven-problems-around-tverbergs-theorem/ .