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 Eurasian Math. J., 2012, Volume 3, Number 2, Pages 21–30 (Mi emj84)

Monotone path-connectedness of $R$-weakly convex sets in spaces with linear ball embedding

A. R. Alimov

Faculty of Mechanics and Mathematics, M. V. Lomonosov Moscow State University, Moscow, Russia

Abstract: A subset $M$ of a normed linear space $X$ is called $R$-weakly convex ($R>0$) if $(D_R(x,y)\setminus\{x,y\})\cap M\ne\varnothing$ for any $x,y\in M$ satisfying $0<\|x-y\|<2R$. Here, $D_R(x,y)$ is the intersection of all closed balls of radius $R$ containing $x,y$. The paper is concerned with the connectedness of $R$-weakly convex subsets of Banach spaces satisfying the linear ball embedding condition $\mathrm{(BEL)}$ (note that $C(Q)$ and $\ell^1(n)\in\mathrm{(BEL)}$). An $R$-weakly convex subset $M$ of a space $X\in\mathrm{(BEL)}$ is shown to be mconnected (Menger-connected) under the natural condition on the spread of points in $M$. A closed subset $M$ of a finite-dimensional space $X\in\mathrm{(BEL)}$ is shown to be $R$-weakly convex with some $R>0$ if and only if $M$ is a disjoint union of monotone path-connected suns in $X$, the Hausdorff distance between any connected components of $M$ being less than $2R$. In passing we obtain a characterization of three-dimensional spaces with subequilateral unit ball.

Keywords and phrases: Chebyshev set, sun, strict sun, normed linear space, linear ball embedding, interval, span, bar, extreme functional.

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Bibliographic databases:
MSC: 52A30, 41A65, 46B20
Citation: 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
\Bibitem{Ali12} \by A.~R.~Alimov \paper Monotone path-connectedness of $R$-weakly convex sets in spaces with linear ball embedding \jour Eurasian Math. J. \yr 2012 \vol 3 \issue 2 \pages 21--30 \mathnet{http://mi.mathnet.ru/emj84} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3024118} \zmath{https://zbmath.org/?q=an:1269.46009}