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Proceedings of the American Mathematical Society

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

 
 

 

A Krasnoselskiĭ-type theorem involving $ p$-arcs


Author: Jean B. Chan
Journal: Proc. Amer. Math. Soc. 102 (1988), 667-676
MSC: Primary 52A35; Secondary 52A30
DOI: https://doi.org/10.1090/S0002-9939-1988-0929000-4
MathSciNet review: 929000
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Abstract: Let $ p$ be a point in $ {E_2}$. A convex arc joining a pair of distinct points $ x$ and $ y$ in $ {E_2}$ is called a $ p$-arc if it is contained in the simplex with vertices $ x,y$, and $ p$. In this paper, we prove the following Krasnosel'skii-type theorem: Let $ S$ be a compact simply connected set in $ {E_2}$ and let $ p$ be a point not in $ S$. If for each three points $ {x_{1,}}{x_2}$, and $ {x_3}$ of $ S$ there exists at least one point $ y \in S$ such that $ y$ and $ {x_i}\left( {i = 1,2,3} \right)$ can be joined by $ p$-arcs in $ S$, then there exists a point $ k \in S$ such that every point $ x \in S$ can be joined to $ k$ by some $ p$-arc in $ S$.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0929000-4
Keywords: Convex sets, theorem of Krasnosel'skii, $ p$-arcs, convex arcs, starshaped sets
Article copyright: © Copyright 1988 American Mathematical Society

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