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Proceedings of the American Mathematical Society

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

 

 

Separating two compact sets by a parallelotope


Author: Steven R. Lay
Journal: Proc. Amer. Math. Soc. 79 (1980), 279-284
MSC: Primary 52A35
DOI: https://doi.org/10.1090/S0002-9939-1980-0565354-1
MathSciNet review: 565354
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Abstract: Let P and Q be compact subsets of Euclidean n-space. Paul Kirchberger has established that there exists a closed half-space M such that $ P \subset M$ and $ Q \cap M = \emptyset $ iff for each set S consisting of $ n + 2$ or fewer points of $ P \cup Q$, there exists a closed half-space $ {M_S}$ such that $ (P \cap S) \subset {M_S}$ and $ (Q \cap S) \cap {M_S} = \emptyset $. In this paper the problem of replacing the half-spaces by parallelotopes is considered, and the critical number of points in $ P \cup Q$ is shown to be $ n + 1$. Applications of this are drawn to systems of linear inequalities and to Carathéodory's theorem.


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DOI: https://doi.org/10.1090/S0002-9939-1980-0565354-1
Keywords: Separation of compact sets, parallelotope, Kirchberger's theorem, Carathéodory's theorem
Article copyright: © Copyright 1980 American Mathematical Society