A flat strip theorem for Ptolemaic spaces

Abstract

A metric space (X,d) is called Ptolemaic space (PT space) if the inequality |xy||zw|≤|xz||yw|+|xw||yz| holds for each quadruple of points x, y, z and w in X. Here |xy|:=d(x,y) denotes the distance of two points x and y. The main result proven in the paper is the following: If (X,d) is a PT space which is also a geodesic space (i.e., for each pair x,y there exists a geodesic connecting x and y) and if X is moreover homeomorphic to ℝ×[0,1] then X is isometric to a flat strip ℝ×[0,a]⊂ℝ 2 with its Euclidean metric. The authors also give a new short proof of the fact that a proper geodesic PT space is always strictly distance convex. (See also Th. Foertsch and V. Schroeder [Trans. Am. Math. Soc. 363, No. 6, 2891–2906 (2011; Zbl 1220.53092)]). This also indicates a positive answer to the open question if every proper geodesic PT space is also a CAT(0)-space.