The Crossing Tverberg Theorem.
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| Title: | The Crossing Tverberg Theorem. |
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| 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] |
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| Database: | Engineering Source |
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| 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] |
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| ISSN: | 01795376 |
| DOI: | 10.1007/s00454-023-00532-x |
