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

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.