Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 52A30, 52A35

Retrieve articles in all journals with MSC: 52A30, 52A35

Additional Information

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