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 Fundam. Prikl. Mat., 2012, Volume 17, Issue 1, Pages 23–32 (Mi fpm1387)

Monotone path-connectedness of $R$-weakly convex sets in the space $C(Q)$

A. R. Alimov

M. V. Lomonosov Moscow State University

Abstract: A subset $M$ of a normed linear space $X$ is said to be $R$-weakly convex ($R>0$ is fixed) if the intersection $(D_R(x,y)\setminus\{x,y\})\cap M$ is nonempty for all $x,y\in M$, $0<\|x-y\|<2R$. Here $D_R(x,y)$ is the intersection of all the balls of radius $R$ that contain $x,y$. The paper is concerned with connectedness of $R$-weakly convex sets in $C(Q)$-spaces. It will be shown that any $R$-weakly convex subset $M$ of $C(Q)$ is locally $\mathrm m$-connected (locally Menger-connected) and each connected component of a boundedly compact $R$-weakly convex subset $M$ of $C(Q)$ is monotone path-connected and is a sun in $C(Q)$. Also, we show that a boundedly compact subset $M$ of $C(Q)$ is $R$-weakly convex for some $R>0$ if and only if $M$ is a disjoint union of monotonically path-connected suns in $C(Q)$, the Hausdorff distance between each pair of the components of $M$ being at least $2R$.

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English version:
Journal of Mathematical Sciences (New York), 2012, 185:3, 360–366

Document Type: Article
UDC: 517.982.252+517.982.256

Citation: A. R. Alimov, “Monotone path-connectedness of $R$-weakly convex sets in the space $C(Q)$”, Fundam. Prikl. Mat., 17:1 (2012), 23–32; J. Math. Sci., 185:3 (2012), 360–366

Citation in format AMSBIB
\Bibitem{Ali12} \by A.~R.~Alimov \paper Monotone path-connectedness of $R$-weakly convex sets in the space $C(Q)$ \jour Fundam. Prikl. Mat. \yr 2012 \vol 17 \issue 1 \pages 23--32 \mathnet{http://mi.mathnet.ru/fpm1387} \transl \jour J. Math. Sci. \yr 2012 \vol 185 \issue 3 \pages 360--366 \crossref{https://doi.org/10.1007/s10958-012-0920-2} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84866329641} 

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• http://mi.mathnet.ru/eng/fpm/v17/i1/p23

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. R. Alimov, “Monotone path-connectedness of $R$-weakly convex sets in spaces with linear ball embedding”, Eurasian Math. J., 3:2 (2012), 21–30
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