Monotone path-connectedness of R-weakly convex sets in spaces with linear ball embedding

Abstract

A subset \(M\) of a normed linear space \(X\) is called \(R\)-weakly convex (\(R > 0\)) if \((D_R(x, y) \setminus \{x, y\}) \cap M \neq \emptyset\) for any \(x, y \in M\) satisfying \(0 < \|x-y\| < 2R\). 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 (BEL) (note that \(C(Q)\) and \(\ell_1^n \in (BEL)\)). An \(R\)-weakly convex subset \(M\) of a space \(X \in (BEL)\) is shown to be \(m\)-connected (Menger-connected) under the natural condition on the spread of points in \(M\). A closed subset \(M\) of a finite-dimensional space \(X \in (BEL)\) 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.