Аннотация:A subset $M$ of a~normed linear space~$X$ is called
$R$-weakly convex ($R>0$) if $(D_R(x,y)\setminus \{x,y\}) \cap M\ne\emptyset$
for any $x,y\in M$, $ 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 $\ABE$
(note that $C(Q)$ and~$\ell^1(n)\in \ABE$).
An $R$-weakly convex subset~$M$ of a~space
$X\in \ABE$\enskip is shown to be $\mcc$-connected (Menger-connected) under the natural condition
on the spread of points in~$M$. A~closed subset~$M$ of
a~finite-dimensional space $X\in \ABE$ 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.