The crossing Tverberg theorem

article The crossing Tverberg theorem published yes Radoslav Fulek author 39F3FFE4-F248-11E8-B48F-1D18A9856A870000-0001-8485-1774 Bernd Gärtner author Andrey Kupavskii author Pavel Valtr author Uli Wagner author 36690CA2-F248-11E8-B48F-1D18A9856A870000-0002-1494-0568 UlWa department Eliminating intersections in drawings of graphs project 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. Springer Nature2024 eng Discrete and Computational Geometry 0179-5376 1432-0444 1812.04911 00103854650000110.1007/s00454-023-00532-x 72831-848 https://research-explorer.ista.ac.at/record/6647 Fulek, Radoslav, et al. “The Crossing Tverberg Theorem.” Discrete and Computational Geometry, vol. 72, Springer Nature, 2024, pp. 831–48, doi:<a href="https://doi.org/10.1007/s00454-023-00532-x">10.1007/s00454-023-00532-x</a>. Fulek R, Gärtner B, Kupavskii A, Valtr P, Wagner U. The crossing Tverberg theorem. Discrete and Computational Geometry. 2024;72:831-848. doi:<a href="https://doi.org/10.1007/s00454-023-00532-x">10.1007/s00454-023-00532-x</a> R. Fulek, B. Gärtner, A. Kupavskii, P. Valtr, U. Wagner, Discrete and Computational Geometry 72 (2024) 831–848. Fulek R, Gärtner B, Kupavskii A, Valtr P, Wagner U. 2024. The crossing Tverberg theorem. Discrete and Computational Geometry. 72, 831–848. Fulek, R., Gärtner, B., Kupavskii, A., Valtr, P., & Wagner, U. (2024). The crossing Tverberg theorem. Discrete and Computational Geometry. Springer Nature. <a href="https://doi.org/10.1007/s00454-023-00532-x">https://doi.org/10.1007/s00454-023-00532-x</a> R. Fulek, B. Gärtner, A. Kupavskii, P. Valtr, and U. Wagner, “The crossing Tverberg theorem,” Discrete and Computational Geometry, vol. 72. Springer Nature, pp. 831–848, 2024. Fulek, Radoslav, Bernd Gärtner, Andrey Kupavskii, Pavel Valtr, and Uli Wagner. “The Crossing Tverberg Theorem.” Discrete and Computational Geometry. Springer Nature, 2024. <a href="https://doi.org/10.1007/s00454-023-00532-x">https://doi.org/10.1007/s00454-023-00532-x</a>. 139742023-08-06T22:01:12Z2025-04-14T13:52:36Z