The Bárány-Kalai conjecture for certain families of polytopes
arxiv(2024)
摘要
Bárány and Kalai conjectured the following generalization of Tverberg's
theorem: if f is a linear function from an m-dimensional polytope P to
ℝ^d 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.
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