139,919 research outputs found

    Quantitative Tverberg theorems over lattices and other discrete sets

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    This paper presents a new variation of Tverberg's theorem. Given a discrete set SS of RdR^d, we study the number of points of SS needed to guarantee the existence of an mm-partition of the points such that the intersection of the mm convex hulls of the parts contains at least kk points of SS. The proofs of the main results require new quantitative versions of Helly's and Carath\'eodory's theorems.Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:1503.0611

    Quantitative Tverberg, Helly, & Carath\'eodory theorems

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    This paper presents sixteen quantitative versions of the classic Tverberg, Helly, & Caratheodory theorems in combinatorial convexity. Our results include measurable or enumerable information in the hypothesis and the conclusion. Typical measurements include the volume, the diameter, or the number of points in a lattice.Comment: 33 page
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