  B\'ar\'any and Kalai conjectured the following generalization of Tverberg's
theorem: if $f$ is a linear function from an $m$-dimensional polytope $P$ to
$\mathbb{R}^d$ and $m \ge (d + 1)(r - 1)$, then there are $r$ pairwise disjoint
faces of $P$ whose images have a point in common. We show that the conjecture
holds for cross polytopes, cyclic polytopes, and more generally for
$(d+1)$-neighborly polytopes. Moreover, we show that for cross polytopes, the
conjecture holds if the map $f$ is assumed to be continuous (but not
necessarily linear), and we give a lower bound on the number of sets of $r$
pairwise disjoint faces whose images under $f$ intersect. We also show that the
conjecture holds for all polytopes when $d=1$ and $f$ is assumed to be
continuous. Finally, when $r$ is prime or large enough with respect to $d$, we
prove that there exists a constant $c=c(d,r)$, depending only on $d$ and $r$,
such that the conjecture holds (with continuous functions) for the polytope
obtained by taking $c$ subdivisions of $P$.
