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

A Krasnoselskiĭ-type theorem for points of local nonconvexity


Author: Marilyn Breen
Journal: Proc. Amer. Math. Soc. 85 (1982), 261-266
MSC: Primary 52A10; Secondary 52A30
MathSciNet review: 652454
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Abstract: Let $ S$ be a compact connected set in $ {R^2}$, $ S$ not convex. Then $ S$ is starshaped if and only if every 3 points of local nonconvexity of $ S$ are clearly visible from a common point of $ S$. For $ k = 1$ or $ k = 2$, dimker $ S \geqslant $ $ k$ if and only if for some $ \in > 0$, every $ f(k) = \max \left\{ {3,6 - 2k} \right\}$ points of local nonconvexity of $ S$ are clearly visible from a common $ k$-dimensional $ \in $neighborhood in $ S$. Each result is best possible.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1982-0652454-2
PII: S 0002-9939(1982)0652454-2
Keywords: Starshaped sets, convex kernel, points of local nonconvexity
Article copyright: © Copyright 1982 American Mathematical Society