A metric space \((X, d)\) is called Ptolemaic space (PT space) if the inequality \[ |x \, y| |z \, w| \;\; \leq \;\; |x \, z| |y \, w| + |x \, w| |y \, z| \] holds for each quadruple of points \(x\), \(y\), \(z\) and \(w\) in \(X\). Here \(|x \, y| := d(x, y)\) 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 \({\mathbb R}\times [0,1]\) then \(X\) is isometric to a flat strip \({\mathbb R}\times [0,a] \subset {\mathbb R}^2\) 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 [T. Foertsch and the second author, Trans. Am. Math. Soc. 363, No. 6, 2891–2906 (2011; Zbl 1220.53092)]). This also indicates a positive answer to the open question whether every proper geodesic PT space is also a CAT(0)-space.