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

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 $2R$.