On local properties of spaces implying monotone path-connectedness of suns

The seminal A. L. Brown’s theorem asserts that in a finite-dimensional (BM)-space any sun is Menger connected, and in view of one earlier result of the author, is monotone path-connected. There is a well-known characterization of normed spaces of dimension 3 and 4 in which every Chebyshev set is convex, is that every exposed point of the unit sphere must be smooth. It turns out that, for a given set M, one can verify its convexity by testing for smoothness not all exposed points of the sphere, but only the so-called M-acting points of the sphere, i.e., the points such that an appropriate translation of a homothetic copy of the unit ball “touches” M with an “analogue” of this point. In the present paper, analogous to this result, we establish monotone path-connectedness of sets. Namely, we show that a sun M of a finite-dimensional normed space is monotone path-connected whenever any M-acting point of the unit sphere is a (BM)-point. We also give a characterization of three-dimensional polyhedral spaces in which every sun is monotone path-connected.