{"publisher":"Springer Nature","scopus_import":"1","oa_version":"Preprint","oa":1,"date_created":"2023-08-06T22:01:12Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","title":"The crossing Tverberg theorem","quality_controlled":"1","status":"public","related_material":{"record":[{"relation":"earlier_version","id":"6647","status":"public"}]},"article_processing_charge":"No","month":"07","date_updated":"2023-12-13T12:03:35Z","type":"journal_article","day":"27","department":[{"_id":"UlWa"}],"abstract":[{"lang":"eng","text":"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 Rd, one can find a partition X=X1∪⋯∪Xr of X, such that the convex hulls of the Xi, i=1,…,r, all share a common point. In this paper, we prove a trengthening 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 Rd, d≥2."}],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.1812.04911"}],"date_published":"2023-07-27T00:00:00Z","publication_status":"epub_ahead","year":"2023","project":[{"call_identifier":"FWF","grant_number":"M02281","_id":"261FA626-B435-11E9-9278-68D0E5697425","name":"Eliminating intersections in drawings of graphs"}],"acknowledgement":"Part of the research leading to this paper was done during the 16th Gremo Workshop on Open Problems (GWOP), Waltensburg, Switzerland, June 12–16, 2018. We thank Patrick Schnider for suggesting the problem, and Stefan Felsner, Malte Milatz, and Emo Welzl for fruitful discussions during the workshop. We also thank Stefan Felsner and Manfred Scheucher for finding, communicating the example from Sect. 3.3, and the kind permission to include their visualization of the point set. We thank Dömötör Pálvölgyi, the SoCG reviewers, and DCG reviewers for various helpful comments.\r\nR. Fulek gratefully acknowledges support from Austrian Science Fund (FWF), Project  M2281-N35. A. Kupavskii was supported by the Advanced Postdoc.Mobility Grant no. P300P2_177839 of the Swiss National Science Foundation. Research by P. Valtr was supported by the Grant no. 18-19158 S of the Czech Science Foundation (GAČR).","isi":1,"external_id":{"arxiv":["1812.04911"],"isi":["001038546500001"]},"publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"publication":"Discrete and Computational Geometry","author":[{"id":"39F3FFE4-F248-11E8-B48F-1D18A9856A87","full_name":"Fulek, Radoslav","first_name":"Radoslav","last_name":"Fulek","orcid":"0000-0001-8485-1774"},{"full_name":"Gärtner, Bernd","first_name":"Bernd","last_name":"Gärtner"},{"full_name":"Kupavskii, Andrey","last_name":"Kupavskii","first_name":"Andrey"},{"full_name":"Valtr, Pavel","last_name":"Valtr","first_name":"Pavel"},{"orcid":"0000-0002-1494-0568","last_name":"Wagner","first_name":"Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","full_name":"Wagner, Uli"}],"doi":"10.1007/s00454-023-00532-x","_id":"13974","language":[{"iso":"eng"}],"citation":{"mla":"Fulek, Radoslav, et al. “The Crossing Tverberg Theorem.” <i>Discrete and Computational Geometry</i>, Springer Nature, 2023, doi:<a href=\"https://doi.org/10.1007/s00454-023-00532-x\">10.1007/s00454-023-00532-x</a>.","ieee":"R. Fulek, B. Gärtner, A. Kupavskii, P. Valtr, and U. Wagner, “The crossing Tverberg theorem,” <i>Discrete and Computational Geometry</i>. Springer Nature, 2023.","apa":"Fulek, R., Gärtner, B., Kupavskii, A., Valtr, P., &#38; Wagner, U. (2023). The crossing Tverberg theorem. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-023-00532-x\">https://doi.org/10.1007/s00454-023-00532-x</a>","ista":"Fulek R, Gärtner B, Kupavskii A, Valtr P, Wagner U. 2023. The crossing Tverberg theorem. Discrete and Computational Geometry.","chicago":"Fulek, Radoslav, Bernd Gärtner, Andrey Kupavskii, Pavel Valtr, and Uli Wagner. “The Crossing Tverberg Theorem.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s00454-023-00532-x\">https://doi.org/10.1007/s00454-023-00532-x</a>.","ama":"Fulek R, Gärtner B, Kupavskii A, Valtr P, Wagner U. The crossing Tverberg theorem. <i>Discrete and Computational Geometry</i>. 2023. doi:<a href=\"https://doi.org/10.1007/s00454-023-00532-x\">10.1007/s00454-023-00532-x</a>","short":"R. Fulek, B. Gärtner, A. Kupavskii, P. Valtr, U. Wagner, Discrete and Computational Geometry (2023)."},"article_type":"original"}