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Proceedings of the American Mathematical Society

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

 

 

A weak Krasnoselskiĭ theorem in $ {\bf R}\sp d$


Author: Marilyn Breen
Journal: Proc. Amer. Math. Soc. 104 (1988), 558-562
MSC: Primary 52A35; Secondary 52A30
DOI: https://doi.org/10.1090/S0002-9939-1988-0962828-3
MathSciNet review: 962828
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Abstract: Let $ S$ be a compact, locally starshaped set in $ {R^d}$, and let $ k$ be a fixed integer, $ 0 \leq k \leq d$. If every $ d - k + 1$ points of $ S$ are clearly visible via $ S$ from a common point, then for every $ k$-flat $ F'$ there exists a translate $ F$ of $ F'$ such that the following holds:

To each point $ s_0$ in $ S \sim F$ there correspond a point $ {s_m}$ in $ F$ and a polygonal path $ \bigcup \left\{ {[{s_{i - 1}},{s_i}]:1 \leq i \leq m} \right\}$ in $ S \cap \operatorname{aff} ({s_0} \cup F)$ with $ \operatorname{dist} ({s_i},F) < \operatorname{dist} ({s_{i - 1}},F),1 \leq i \leq m$.

If $ k = 0$ or $ k = d - 1$, then each point of $ S$ sees via $ S$ some point of $ F$. Moreover, if $ k = 1$, then $ F$ can be chosen so that $ F \cap S$ is convex.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0962828-3
Article copyright: © Copyright 1988 American Mathematical Society