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Quantitative Tverberg Theorems Over Lattices and Other Discrete Sets

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Abstract

This paper presents a new variation of Tverberg’s theorem. Given a discrete set S of \(\mathbb {R}^{d}\), we study the number of points of S needed to guarantee the existence of an m-partition of the points such that the intersection of the m convex hulls of the parts contains at least k points of S. The proofs of the main results require new quantitative versions of Helly’s and Carathéodory’s theorems.

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Acknowledgements

We are grateful to G. Averkov, I. Bárány, A. Barvinok, F. Frick, B. González Merino, A. Holmsen, J. Pach, S. Weltge, and G.M. Ziegler for their comments and suggestions. This work was partially supported by the Institute for Mathematics and its Applications (IMA) in Minneapolis, MN funded by the National Science Foundation (NSF). The authors are grateful for the wonderful working environment that led to this paper. The research of De Loera and La Haye was also supported first by a UC MEXUS grant and later by an NSA grant. De Loera was also supported by NSF Grant DMS-1522158. Rolnick was additionally supported by NSF Grants DMS-1321794 and 1122374.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Davis, Davis, CA, 95616, USA

    Jesus A. De Loera & Reuben N. La Haye

  2. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    David Rolnick

  3. Department of Mathematics, Northeastern University, Boston, MA, 02115, USA

    Pablo Soberón

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  1. Jesus A. De Loera
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  2. Reuben N. La Haye
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  3. David Rolnick
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  4. Pablo Soberón
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Correspondence to Pablo Soberón.

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Editor in Charge: Günter M. Ziegler

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De Loera, J.A., La Haye, R.N., Rolnick, D. et al. Quantitative Tverberg Theorems Over Lattices and Other Discrete Sets. Discrete Comput Geom 58, 435–448 (2017). https://doi.org/10.1007/s00454-016-9858-3

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