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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Clear visibility and the dimension of kernels of starshaped sets
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by Marilyn Breen PDF
Proc. Amer. Math. Soc. 85 (1982), 414-418 Request permission

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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Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 85 (1982), 414-418
  • MSC: Primary 52A30; Secondary 52A35
  • DOI: https://doi.org/10.1090/S0002-9939-1982-0656115-5
  • MathSciNet review: 656115