Quantitative Steinitz Theorem and Polarity.
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| Title: | Quantitative Steinitz Theorem and Polarity. |
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| 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] |
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| Database: | Engineering Source |
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| 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] |
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| ISSN: | 01795376 |
| DOI: | 10.1007/s00454-025-00743-4 |
