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

Breen, Marilyn

doi:10.1090/S0002-9939-1982-0652454-2

A Krasnosel′skiĭ-type theorem for points of local nonconvexity
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Proc. Amer. Math. Soc. 85 (1982), 261-266 Request permission

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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Local convexity and

sets

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