Birch's conjecture for partitions with intersecting convex hulls

Let $r$ and $d$ be positive integers, and let $N$ denote the number of points under consideration. Birch's conjecture. Any $(r-1)(d+1)+1$ points in $\mathbb{R}^{d}$ can be partitioned into $N$ subsets whose convex hulls have a common point. Birch's conjecture was proved by Helge Tverberg and is now known as the Tverberg theorem.