Quantitative Steinitz theorem: a spherical version

Abstract Steinitz’s theorem states that if the origin belongs to the interior of the convex hull of a set $$Q \subset {\mathbb {R}}^d,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> then there are at most 2 d points $$Q^\prime $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Q</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> 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 $$Q^\prime $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Q</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> is bounded from below. In the present note, we show that a Euclidean result of this kind implies a corresponding spherical version.