The B\'ar\'any-Kalai conjecture for certain families of polytopes
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 $\mathbb{R}^d$ 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 …