Abstract
We will prove the following generalisation of Tverberg’s Theorem: given a set S⊂ℝd of (r+1)(k−1)(d+1)+1 points, there is a partition of S in k sets A 1,A 2,…,A k such that for any C⊂S of at most r points, the convex hulls of A 1\C,A 2\C,…,A k \C are intersecting. This was conjectured first by Natalia García-Colín (Ph.D. thesis, University College of London, 2007).
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Soberón, P., Strausz, R. A Generalisation of Tverberg’s Theorem. Discrete Comput Geom 47, 455–460 (2012). https://doi.org/10.1007/s00454-011-9379-z
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DOI: https://doi.org/10.1007/s00454-011-9379-z