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The Crossing Tverberg Theorem.

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Title: The Crossing Tverberg Theorem.
Authors: Fulek, Radoslav1 (AUTHOR) radoslav.fulek@gmail.com, Gärtner, Bernd2 (AUTHOR), Kupavskii, Andrey3,4 (AUTHOR), Valtr, Pavel5 (AUTHOR), Wagner, Uli6 (AUTHOR)
Source: Discrete & Computational Geometry. Sep2024, Vol. 72 Issue 2, p831-848. 18p.
Subjects: Discrete geometry, Intersection numbers, Generalization
Abstract: The Tverberg theorem is one of the cornerstones of discrete geometry. It states that, given a set X of at least (d + 1) (r - 1) + 1 points in R d , one can find a partition X = X 1 ∪ ⋯ ∪ X r of X, such that the convex hulls of the X i , i = 1 , ... , r , all share a common point. In this paper, we prove a strengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. As a concrete application, we show that any n points in the plane in general position span ⌊ n / 3 ⌋ vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Álvarez-Rebollar et al. guarantees ⌊ n / 6 ⌋ pairwise crossing triangles. Our result generalizes to a result about simplices in R d , d ≥ 2 . [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: The Tverberg theorem is one of the cornerstones of discrete geometry. It states that, given a set X of at least (d + 1) (r - 1) + 1 points in R d , one can find a partition X = X 1 ∪ ⋯ ∪ X r of X, such that the convex hulls of the X i , i = 1 , ... , r , all share a common point. In this paper, we prove a strengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. As a concrete application, we show that any n points in the plane in general position span ⌊ n / 3 ⌋ vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Álvarez-Rebollar et al. guarantees ⌊ n / 6 ⌋ pairwise crossing triangles. Our result generalizes to a result about simplices in R d , d ≥ 2 . [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/s00454-023-00532-x
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      – Code: eng
        Text: English
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              M: 09
              Text: Sep2024
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