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Proceedings of the American Mathematical Society

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

 

 

Separation by cylindrical surfaces


Author: Steven R. Lay
Journal: Proc. Amer. Math. Soc. 36 (1972), 224-228
MSC: Primary 52A35
DOI: https://doi.org/10.1090/S0002-9939-1972-0310767-1
MathSciNet review: 0310767
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Abstract: C. Carathéodory has established that two compact sets P and Q in Euclidean n-space can be strictly separated by a hyperplane if each subset of $ n + 1$ or fewer points of Q can be strictly separated from P by a hyperplane. In this paper it is shown that if each subset of k or fewer points of Q can be strictly separated from P by a hyperplane (where k is a fixed integer, $ 1 \leqq k \leqq n$), then there exists a cylinder of an appropriate sort containing P and disjoint from Q.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0310767-1
Keywords: Separation of convex sets, cylindrical surfaces, Carathéodory theorem, Helly theorem
Article copyright: © Copyright 1972 American Mathematical Society