Quantitative Tverberg Theorems Over Lattices and Other Discrete Sets

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...

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Bibliographic Details
Published in:Discrete & computational geometry 2017-09, Vol.58 (2), p.435-448
Main Authors: De Loera, Jesus A., La Haye, Reuben N., Rolnick, David, Soberón, Pablo
Format: Article
Language:English
Subjects:
Combinatorics
Computational Mathematics and Numerical Analysis
Convexity
Hulls
Lattices
Mathematics
Mathematics and Statistics
Theorems
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Summary: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 Carathéodory’s theorems.
ISSN:0179-5376
1432-0444
DOI:10.1007/s00454-016-9858-3