Quantitative Steinitz theorem: A polynomial bound

Abstract

AbstractThe classical Steinitz theorem states that if the origin belongs to the interior of the convex hull of a set , then there are at most points of whose convex hull contains the origin in the interior. Bárány, Katchalski, and Pach proved the following quantitative version of Steinitz's theorem. Let be a convex polytope in containing the standard Euclidean unit ball . Then there exist at most vertices of whose convex hull satisfies with . They conjectured that holds with a universal constant . We prove , the first polynomial lower bound on . Furthermore, we show that is not greater than .