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Quantitative Steinitz Theorem and Polarity.

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Title: Quantitative Steinitz Theorem and Polarity.
Authors: Ivanov, Grigory1 (AUTHOR) grimivanov@gmail.com
Source: Discrete & Computational Geometry. Jun2026, Vol. 75 Issue 4, p1405-1413. 9p.
Subjects: Mathematical bounds, Polyhedra, Convex domains, Unit ball (Mathematics), Duality theory (Mathematics)
Abstract: The classical Steinitz theorem asserts that if the origin lies within the interior of the convex hull of a set S ⊂ R d , then there are at most 2d points in S whose convex hull contains the origin within its interior. Bárány, Katchalski, and Pach established a quantitative version of Steinitz's theorem, showing that for a convex polytope Q in R d containing the standard Euclidean unit ball B d , there exist at most 2d vertices of Q whose convex hull Q ′ satisfies r B d ⊂ Q ′ with r ≥ d - 2 d . Recently, Márton Naszódi and the author derived a polynomial bound on r. This paper aims to establish a bound on r based on the number of vertices of Q. In other words, we demonstrate an effective method to remove several points from the original set Q without significantly altering the bound on r. Specifically, if the number of vertices of Q scales linearly with the dimension, i.e., α d , then one can select 2d vertices such that r ≥ 1 5 α d . The proof relies on a polarity trick, which may be of independent interest: we demonstrate the existence of a point c in the interior of a convex polytope P ⊂ R d such that the vertices of the polar polytope (P - c) ∘ sum up to zero. [ABSTRACT FROM AUTHOR]
Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="DE" term="%22Mathematical+bounds%22">Mathematical bounds</searchLink><br /><searchLink fieldCode="DE" term="%22Polyhedra%22">Polyhedra</searchLink><br /><searchLink fieldCode="DE" term="%22Convex+domains%22">Convex domains</searchLink><br /><searchLink fieldCode="DE" term="%22Unit+ball+%28Mathematics%29%22">Unit ball (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Duality+theory+%28Mathematics%29%22">Duality theory (Mathematics)</searchLink>
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  Data: The classical Steinitz theorem asserts that if the origin lies within the interior of the convex hull of a set S ⊂ R d , then there are at most 2d points in S whose convex hull contains the origin within its interior. Bárány, Katchalski, and Pach established a quantitative version of Steinitz's theorem, showing that for a convex polytope Q in R d containing the standard Euclidean unit ball B d , there exist at most 2d vertices of Q whose convex hull Q ′ satisfies r B d ⊂ Q ′ with r ≥ d - 2 d . Recently, Márton Naszódi and the author derived a polynomial bound on r. This paper aims to establish a bound on r based on the number of vertices of Q. In other words, we demonstrate an effective method to remove several points from the original set Q without significantly altering the bound on r. Specifically, if the number of vertices of Q scales linearly with the dimension, i.e., α d , then one can select 2d vertices such that r ≥ 1 5 α d . The proof relies on a polarity trick, which may be of independent interest: we demonstrate the existence of a point c in the interior of a convex polytope P ⊂ R d such that the vertices of the polar polytope (P - c) ∘ sum up to zero. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1007/s00454-025-00743-4
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        Text: English
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        Type: general
      – SubjectFull: Polyhedra
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      – SubjectFull: Convex domains
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              Text: Jun2026
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