Quantitative Steinitz theorem: A spherical version

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 $Q^{\mathrm{\prime }}$ 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 $Q^{\mathrm{\prime }}$ is bounded from below. In the present note, we show that a Euclidean result of this kind implies a corresponding spherical version.