Abstract
The classical 1966 theorem of Tverberg with its numerous variations was and still is a motivating force behind many important developments in convex and computational geometry as well as a testing ground for methods from equivariant algebraic topology. In 2018, Bárány and Soberón presented a new variation, the “Tverberg plus minus theorem.” In this paper, we give a new proof of the Tverberg plus minus theorem, by using a projective transformation. The same tool allows us to derive plus minus analogues of all known affine Tverberg type results. In particular, we prove a plus minus analogue of the optimal colored Tverberg theorem.
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References
Bárány, I., Blagojević, P.V.M., Ziegler, G.M.: Tverberg’s theorem at 50: extensions and counterexamples. Notices Am. Math. Soc. 63(7), 732–739 (2016)
Bárány, I., Larman, D.G.: A colored version of Tverberg’s theorem. J. Lond. Math. Soc. 45(2), 314–320 (1992)
Bárány, I., Soberón, P.: Tverberg plus minus. Discrete Comput. Geom. 60(3), 588–598 (2018)
Bárány, I., Soberón, P.: Tverberg’s theorem is 50 years old: a survey. Bull. Am. Math. Soc. 55(4), 459–492 (2018)
Blagojević, P.V.M., Matschke, B., Ziegler, G.M.: Optimal bounds for a colorful Tverberg–Vrećica type problem. Adv. Math. 226(6), 5198–5215 (2011)
Blagojević, P.V.M., Matschke, B., Ziegler, G.M.: Optimal bounds for the colored Tverberg problem. J. Eur. Math. Soc. (JEMS) 17(4), 739–754 (2015)
Blagojević, P.V.M., Ziegler, G.M.: Beyond the Borsuk–Ulam theorem: the topological Tverberg story. In: Loebl, M., Nešetřil, J., Thomas, R. (eds.) A Journey Through Discrete Mathematics: A Tribute to Jiří Matoušek, pp. 273–341. Springer, Cham (2017)
De Loera, J.A., Hogan, T.A., Meunier, F., Mustafa, N.: Tverberg theorems over discrete sets of points. arXiv:1803.01816 (2019)
Matoušek, J.: Using the Borsuk–Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry. Universitext. Springer, Heidelberg (2003). Second corrected printing (2008)
Tverberg, H.: A generalization of Radon’s theorem. J. Lond. Math. Soc. 41, 123–128 (1966)
Ziegler, G.M.: $3N$ colored points in a plane. Notices Am. Math. Soc. 58(4), 550–557 (2011)
Acknowledgements
We are grateful to Imre Bárány, Florian Frick, and Pablo Soberón for useful discussions and comments. Furthermore, we thank the the referees, Peter Landweber, and Pablo Soberón for careful reading of the manuscript, valuable suggestions and corrections.
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Dedicated to the memory of Branko Grünbaum.
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The research by Pavle V. M. Blagojević leading to these results has received funding from DFG via the Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics,” and the Grant ON 174024 of the Serbian Ministry of Education and Science. The research by Günter M. Ziegler leading to these results has funding from DFG via the Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics.”
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Blagojević, P.V.M., Ziegler, G.M. Plus Minus Analogues for Affine Tverberg Type Results. Discrete Comput Geom 64, 535–541 (2020). https://doi.org/10.1007/s00454-019-00120-y
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DOI: https://doi.org/10.1007/s00454-019-00120-y