A flat strip theorem for ptolemaic spaces
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Mathematische Zeitschrift
A flat strip theorem for ptolemaic spaces Renlong Miao · Viktor Schroeder
Received: 4 April 2012 / Accepted: 10 May 2012 / Published online: 27 September 2012 © Springer-Verlag Berlin Heidelberg 2012
1 Main result and motivation A metric space (X, d) is called ptolemaic or short a PT space, if for all quadruples of points x, y, z, w ∈ X the Ptolemy inequality |x y| |zw| ≤ |x z| |yw| + |xw| |yz|
(1)
holds, where |x y| denotes the distance d(x, y). We prove a flat strip theorem for geodesic ptolemaic spaces. Two unit speed geodesic lines c0 , c1 : R → X are called parallel, if their distance is sublinear, i.e. if limt→∞ 1t d(c0 (t), c1 (t)) = limt→−∞ 1t d(c0 (t), c1 (t)) = 0. Theorem 1.1 Let X be a geodesic PT space which is homeomorphic to R × [0, 1], such that the boundary curves are parallel geodesic lines, then X is isometric to a flat strip R × [0, a] ⊂ R2 with its euclidean metric. We became interested in ptolemaic metric spaces because of their relation to the geometry of the boundary at infinity of CAT(−1) spaces (compare [3,6]). We therefore think that these spaces have the right to be investigated carefully. Our paper is a contribution to the following question Q: Are proper geodesic ptolemaic spaces CAT(0)-spaces? We give a short discussion of this question at the end of the paper in Sect. 5. Main ingredients of our proof is a theorem of Hitzelberger and Lytchak [8] about isometric embeddings of geodesic spaces into Banach spaces and the Theorem of Schoenberg [10] characterizing inner product spaces by the PT inequality. Finally we thank the referee for the detailed comments. R. Miao · V. Schroeder (B) Institut für Mathematik, Universität Zürich, Winterthurer Strasse 190, 8057 Zurich, Switzerland e-mail: [email protected] R. Miao e-mail: [email protected]
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2 Preliminaries In this section we collect the most important basic facts about geodesic PT spaces which we will need in our arguments. If we do not provide proofs in this section, these can be found in [5,7]. Let X be a metric space. By |x y| we denote the distance between points x, y ∈ X . We will always parametrize geodesics proportionally to arclength. Thus a geodesic in X is a map c : I → X with |c(t)c(s)| = λ|t − s| for all s, t ∈ I and some constant λ ≥ 0. A metric space is called geodesic if every pair of points can be joined by a geodesic. In addition we will use the following convention in this paper. If a geodesic is parametrized on [0, ∞) or on R, the parametrization is always by arclength. A geodesic c : [0, ∞) → X is called a ray, a geodesic c : R → X is called a line. In the sequel X will always denote a geodesic metric space. For x, y ∈ X we denote by m(x, y) = {z ∈ X | |x z| = |zy| = 21 |x y|} the set of midpoints of x and y. A subset C ⊂ X is convex, if for x, y ∈ C also m(x, y) ⊂ C. A function f : X → R is convex (resp. affine), if for all geodesics c : I → X the map f ◦ c : I → R is convex (resp. affine). The space X is called distance convex if for all p ∈
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