Metric characterizations of Euclidean spaces
Gordon Berg · Pacific Journal of Mathematics · 1973
In a metric space an arc which is isometric to a real interval is called a segment.In this paper it is shown that, for 1 ^ n ^ 3, %-dimensional Euclidean space (E n ) is topologically characterized, among locally compact, ^-dimensional spaces, by admitting a metric with the following properties:(1) every two points of the space are endpoints of a unique segment, (2) if two segments have an endpoint and one other point in common then one is contained in the other and (3) every segment can be extended, at either end, to a larger segment.This follows from the more general result that, for 1 ^ n ^ 3, a locally compact, ^-dimensional space which admits a metric with properties (1) and ( 2) is homeomorphic to an ^-manifold lying between the closed n-ball and its interior.Property (1) suffices to characterize E* 9 for n = 1 or 2, among locally compact, locally homogeneous, ^-dimensional spaces.For n > 3, properties (1), (2), and (3) characterize E n among locally compact, ^-dimensional spaces that contain a homeomorph of an n-ball.COROLLARY 2.2.If (X, d) is a locally compact, strongly convex metric space, then the segment between two points is unique and contains