Quantitative Tverberg theorems over lattices and other discrete sets

J. A. De Loera; R. N. La Haye; D. Rolnick; P. Soberón

Quantitative Tverberg theorems over lattices and other discrete sets

Metric Geometry 2016-03-21 v2 Combinatorics

Authors: J. A. De Loera , R. N. La Haye , D. Rolnick , P. Soberón

Abstract

This paper presents a new variation of Tverberg's theorem. Given a discrete set $S$ of $R^d$ , 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 Carath\'eodory's theorems.

Keywords

Tverberg's theorem discrete sets lattices convex hulls quantitative geometry Helly's theorem

Cite

@article{arxiv.1603.05525,
  title  = {Quantitative Tverberg theorems over lattices and other discrete sets},
  author = {J. A. De Loera and R. N. La Haye and D. Rolnick and P. Soberón},
  journal= {arXiv preprint arXiv:1603.05525},
  year   = {2016}
}

Comments

16 pages. arXiv admin note: substantial text overlap with arXiv:1503.06116

Quantitative Tverberg theorems over lattices and other discrete sets

Abstract

Keywords

Cite

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