  Tverberg's theorem is one of the cornerstones of discrete geometry. It states
that, given a set $X$ of at least $(d+1)(r-1)+1$ points in $\mathbb R^d$, one
can find a partition $X=X_1\cup \ldots \cup X_r$ of $X$, such that the convex
hulls of the $X_i$, $i=1,\ldots,r$, all share a common point. In this paper, we
prove a strengthening of this theorem that guarantees a partition which, in
addition to the above, has the property that the boundaries of full-dimensional
convex hulls have pairwise nonempty intersections. Possible generalizations and
algorithmic aspects are also discussed.
  As a concrete application, we show that any $n$ points in the plane in
general position span $\lfloor n/3\rfloor$ vertex-disjoint triangles that are
pairwise crossing, meaning that their boundaries have pairwise nonempty
intersections; this number is clearly best possible. A previous result of
Rebollar et al.\ guarantees $\lfloor n/6\rfloor$ pairwise crossing triangles.
Our result generalizes to a result about simplices in $\mathbb R^d,d\ge2$.
