Monotone path-connectedness and solarity of Menger-connected sets

Abstract

A boundedly compact (boundedly weakly compact) m-connected (Menger-connected) set is shown to be monotone path-\allowbreak connected and is a sun in a broad class of Banach spaces (in particular, in separable spaces). Further, the intersection of a boundedly compact monotone path-\allowbreak connected (m-connected) set with a closed ball is established to be cell-like (trivial shape) and, in particular, acyclic (contractible, in the finite-dimensional case) and a sun. Further, a boundedly weakly compact m-connected set is asserted to be monotone path-\allowbreak connected. In passing, Rainwater--\allowbreak Simons's theorem on weak convergence of sequences is extended to the convergence with respect to the associated norm (in the sense of Brown).