  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.
B\'ar\'any, 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.
