Monotone path-connectedness and solarity of Menger-connected sets in Banach spaces

We prove that every boundedly compact $\operatorname{m}$-connected (Menger-connected) set is monotone path-connected and is a sun in a broad class of Banach spaces (in particular, in separable spaces). We show that the intersection of a boundedly compact monotone path-connected ($\operatorname{m}$-connected) set with a closed ball is cell-like (of trivial shape) and, in particular, acyclic (contractible in the finite-dimensional case) and is a sun. We also prove that every boundedly weakly compact $\operatorname{m}$-connected set is monotone path-connected. In passing, we extend the Rainwater–Simons weak convergence theorem to the case of convergence with respect to the associated norm (in the sense of Brown).