Abstract
We prove a Tverberg type theorem: Given a set \(A \subset \mathbb {R}^d\) in general position with \(|A|=(r-1)(d+1)+1\) and \(k\in \{0,1,\ldots ,r-1\}\), there is a partition of A into r sets \(A_1,\ldots ,A_r\) (where \(|A_j|\le d+1\) for each j) with the following property. There is a unique \(z \in \bigcap _{j=1}^r \mathrm {aff}\,A_j\) and it can be written as an affine combination of the element in \(A_j\): \(z=\sum _{x\in A_j}\alpha (x)x\) for every j and exactly k of the coefficients \(\alpha (x)\) are negative. The case \(k=0\) is Tverberg’s classical theorem.
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Acknowledgements
This material is partly based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the first author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester. Support from Hungarian National Research Grants Nos. K111827 and K116769 is acknowledged. We are also indebted to Attila Pór and Manfred Scheucher for useful discussions, and to an anonymous referee for careful reading and valuable comments.
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Bárány, I., Soberón, P. Tverberg Plus Minus. Discrete Comput Geom 60, 588–598 (2018). https://doi.org/10.1007/s00454-017-9960-1
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DOI: https://doi.org/10.1007/s00454-017-9960-1