Steinitz’s theorem states that if the origin belongs to the interior of the convex hull of a set \(Q \subset {\mathbb {R}}^d,\) then there are at most 2d points \(Q^\prime \) of Q whose convex hull contains the origin in the interior. Bárány, Katchalski and Pach gave a quantitative version, whereby the radius of the ball contained in the convex hull of \(Q^\prime \) is bounded from below. In the present note, we show that a Euclidean result of this kind implies a corresponding spherical version.
