Convex metric spaces with $0$-dimensional midsets

Abstract

Let X be a nontrivial, complete, convex, locally externally convex metric space. Assuming that the midset of each pair of points of X is 0-dimensional and that any nonmaximal metric segment that intersects a midset twice lies in that midset, we show that X is isometric to either the euclidean line El or to a 1-dimensional spherical space S1 a (the circle of radius a in the euclidean plane with the shorter arc metric). The midset of two distinct points a and b in a metric space is defined as the set of all points x in the space for which the distances ax and bx are equal. A metric space X is said to have the weak linear midset property (WLMP) if, for each pair of its distinct points a and b, a nonmaximal (with respect to inclusion) metric segment S belongs to the midset M(a, b) whenever SnM(a, b) contains two points. If, in addition to the WLMP, each midset of a space X is a 0-dimensional set, then we say that X has the 0-dimensional weak linear midset property (0-WLMP). We use the O-WLMP to characterize the euclidean line El and 1-dimensional spherical space Si a among complete, convex, and locally externally convex metric spaces. A 1-dimensional spherical space S1 a is the circle of radius a in the euclidean plane under the shorter arc metric. Berard [2] characterized a topological simple closed curve among convex complete metric spaces with the condition that each midset consist of two points-the double midset property (DMP). We show that Berard's conditions actually yield a characterization of a 1-dimensional spherical