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Proceedings of the American Mathematical Society

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A Krasnosel′skiĭ-type theorem for points of local nonconvexity


Author: Marilyn Breen
Journal: Proc. Amer. Math. Soc. 85 (1982), 261-266
MSC: Primary 52A10; Secondary 52A30
DOI: https://doi.org/10.1090/S0002-9939-1982-0652454-2
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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Keywords: Starshaped sets, convex kernel, points of local nonconvexity
Article copyright: © Copyright 1982 American Mathematical Society