Monotone Path-Connected Boundedly Weakly Compact Set is a Sun

Abstract

We study solar properties of monotone path-connected and Menger-connected boundedly weakly compact sets in normed linear spaces. We show that any Menger-connected boundedly weakly compact subset of a separable normed space is a sun. We also prove that any nonempty monotone path-connected weakly compact subset of normed linear space is a sun. For boundedly weakly compact subsets of the space \(C(Q,\mathbb{R})\), we obtain a criterion of monotone path-connectedness in terms of solarity. Namely, if \(M\) is a nonempty boundedly weakly compact set in \( C(Q)\), then \(M\) is a sun if and only if \(M\) is monotone path-connected. We show that, given a monotone path-connected subset \(M\) of a rotund space \(X\), if \(x\notin M\) has a unique nearest point in \(M\), then \(x\) is a solar point for \(M\). As a corollary, we prove that in a rotund space every proximinal monotonously path-connected set is a Chebyshev sun.