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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Characterizations of the generalized convex kernel


Author: Arthur G. Sparks
Journal: Proc. Amer. Math. Soc. 27 (1971), 563-565
MSC: Primary 52.30
MathSciNet review: 0279692
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Abstract: It is well known that the convex kernel $ K$ of a set $ S$ is the intersection of all maximal convex subsets of $ S$. In this paper it is shown that the $ n$th order kernel of a compact, simply-connected set $ S$ in the plane is an $ {L_n}$ set and is, in fact, the intersection of all maximal $ {L_n}$ subsets of $ S$. Furthermore, it is shown that one does not have to intersect the family of all the maximal $ {L_n}$ subsets to obtain the $ n$th order kernel, but that any subfamily thereof which covers the set is sufficient.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0279692-8
PII: S 0002-9939(1971)0279692-8
Keywords: Convex kernel, generalized convex kernel, $ {L_n}$ sets
Article copyright: © Copyright 1971 American Mathematical Society