Quantitative Steinitz theorem: A polynomial bound

Ivanov, Grigory and Naszódi, Márton (2024) Quantitative Steinitz theorem: A polynomial bound. BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, 56 (2). pp. 796-802. ISSN 0024-6093

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Abstract

The 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 .

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