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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A problem of geometry in $ {\bf R}\sp{n}$


Authors: M. Katchalski and A. Liu
Journal: Proc. Amer. Math. Soc. 75 (1979), 284-288
MSC: Primary 52A35
DOI: https://doi.org/10.1090/S0002-9939-1979-0532152-6
MathSciNet review: 532152
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Abstract: Let $ \mathcal{F}$ be a finite family of at least $ n + 1$ convex sets in the n-dimensional Euclidean space $ {R^n}$. Helly's theorem asserts that if all the $ (n + 1)$-subfamilies of $ \mathcal{F}$ have nonempty intersection, then $ \mathcal{F}$ also has nonempty intersection. The main result in this paper is that if almost all of the $ (n + 1)$-subfamilies of $ \mathcal{F}$ have nonempty intersection, then $ \mathcal{F}$ has a subfamily with nonempty intersection containing almost all of the sets in $ \mathcal{F}$.


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DOI: https://doi.org/10.1090/S0002-9939-1979-0532152-6
Article copyright: © Copyright 1979 American Mathematical Society