The B\'ar\'any-Kalai conjecture for certain families of polytopes

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 Rᵈ and m ≥ (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.