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Quantitative Helly-type theorems

Authors: Imre Bárány, Meir Katchalski and János Pach
Journal: Proc. Amer. Math. Soc. 86 (1982), 109-114
MSC: Primary 52A35
MathSciNet review: 663877
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Abstract: We establish some quantitative versions of Helly's famous theorem on convex sets in Euclidean space. We prove, for instance, that if $ \mathcal{C}$ is any finite family of convex sets in $ {{\mathbf{R}}^d}$, such that the intersection of any $ 2d$ members of $ \mathcal{C}$ has volume at least 1, then the intersection of all members belonging to $ \mathcal{C}$ is of volume $ \geqslant {d^{ - {d^2}}}$. A similar theorem is true for diameter, instead of volume. A quantitative version of Steinitz' Theorem is also proved.

References [Enhancements On Off] (What's this?)

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Article copyright: © Copyright 1982 American Mathematical Society

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