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Proceedings of the American Mathematical Society

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



Clear visibility and the dimension of kernels of starshaped sets

Author: Marilyn Breen
Journal: Proc. Amer. Math. Soc. 85 (1982), 414-418
MSC: Primary 52A30; Secondary 52A35
MathSciNet review: 656115
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Abstract: This paper will use the concept of clearly visible to obtain a Krasnosel’skii-type theorem for the dimension of the kernel of a starshaped set, and the following result will be proved: For each $k$ and $n$, $1 \leqslant k \leqslant n$, let $f(n,n) = n + 1$ and $f(n,k) = 2n$ if $1 \leqslant k \leqslant n - 1$. Let $S$ be a nonempty compact set in ${R^n}$. Then for a $k$ with $1 \leqslant k \leqslant n$, dim ker $S \geqslant k$ if and only if every $f(n,k)$ points of bdry $S$ are clearly visible from a common $k$-dimensional subset of $S$. If $k = 1$ or $k = n$, the result is best possible. Moreover, if $S$ is a compact, connected, nonconvex set in ${R^2}$, then bdry $S$ may be replaced by lnc $S$ in the theorem.

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Keywords: Starshaped sets, convex kernel
Article copyright: © Copyright 1982 American Mathematical Society