The crossing Tverberg theorem

R. Fulek, B. Gärtner, A. Kupavskii, P. Valtr, U. Wagner, in:, 35th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019, p. 38:1-38:13.

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Conference Paper | Published | English
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LIPIcs
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 cup ... cup 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 floor[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 Alvarez-Rebollar et al. guarantees floor[n/6] pairwise crossing triangles. Our result generalizes to a result about simplices in R^d,d >=2.
Publishing Year
Date Published
2019-06-01
Proceedings Title
35th International Symposium on Computational Geometry
Volume
129
Page
38:1-38:13
Conference
SoCG 2019: Symposium on Computational Geometry
Conference Location
Portland, OR, United States
Conference Date
2019-06-18 – 2019-06-21
ISSN
IST-REx-ID

Cite this

Fulek R, Gärtner B, Kupavskii A, Valtr P, Wagner U. The crossing Tverberg theorem. In: 35th International Symposium on Computational Geometry. Vol 129. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2019:38:1-38:13. doi:10.4230/LIPICS.SOCG.2019.38
Fulek, R., Gärtner, B., Kupavskii, A., Valtr, P., & Wagner, U. (2019). The crossing Tverberg theorem. In 35th International Symposium on Computational Geometry (Vol. 129, p. 38:1-38:13). Portland, OR, United States: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPICS.SOCG.2019.38
Fulek, Radoslav, Bernd Gärtner, Andrey Kupavskii, Pavel Valtr, and Uli Wagner. “The Crossing Tverberg Theorem.” In 35th International Symposium on Computational Geometry, 129:38:1-38:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. https://doi.org/10.4230/LIPICS.SOCG.2019.38.
R. Fulek, B. Gärtner, A. Kupavskii, P. Valtr, and U. Wagner, “The crossing Tverberg theorem,” in 35th International Symposium on Computational Geometry, Portland, OR, United States, 2019, vol. 129, p. 38:1-38:13.
Fulek R, Gärtner B, Kupavskii A, Valtr P, Wagner U. 2019. The crossing Tverberg theorem. 35th International Symposium on Computational Geometry. SoCG 2019: Symposium on Computational Geometry, LIPIcs, vol. 129. 38:1-38:13.
Fulek, Radoslav, et al. “The Crossing Tverberg Theorem.” 35th International Symposium on Computational Geometry, vol. 129, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019, p. 38:1-38:13, doi:10.4230/LIPICS.SOCG.2019.38.
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