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The Ptolemaean inequality in the closure of complex hyperbolic planes

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This paper is organized as follows: In the preliminary Section 2 we state well-known facts about the complex hyperbolic plane, its boundary, horospherical coordinates, and the Cygan metric. In Section 3 we prove the Ptolemaean inequality (Theorem 3.1), and finally in Section 4 we prove Ptolemaeus’ theorem. | Turk J Math (2017) 41: 1108 – 1120 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1604-6 Research Article The Ptolemaean inequality in the closure of complex hyperbolic planes 1 2,∗ ¨ Ioannis D. PLATIS1 , Nilg¨ un SONMEZ Department of Mathematics and Applied Mathematics, University of Crete, Heraklion, Crete, Greece 2 Department of Mathematics, Afyon Kocatepe University, Afyonkarahisar, Turkey Received: 02.04.2016 • Accepted/Published Online: 24.10.2016 • Final Version: 28.09.2017 Abstract: We prove the Ptolemaean inequality and Ptolemaeus’ theorem in the closure of complex hyperbolic planes endowed with the Cygan metric. Key words: Complex hyperbolic plane, Cygan metric, Ptolemaean inequality, Ptolemaeus’ theorem 1. Introduction Let (S, ρ) be a metric space. The metric d is called Ptolemaean if for any quadruple of points the Ptolemaean Inequality holds; that is, for any distinct points p1 , p2 , p3 , and p4 ρ(p1 , p3 ) · ρ(p2 , p4 ) ≤ ρ(p1 , p2 ) · ρ(p3 , p4 ) + ρ(p2 , p3 ) · ρ(p4 , p1 ). A subset σ of S is called a Ptolemaean circle if for any four distinct points p1 , p2 , p3 , and p4 in σ such that p1 and p3 separate p2 and p4 we have ρ(p1 , p3 ) · ρ(p2 , p4 ) = ρ(p1 , p2 ) · ρ(p3 , p4 ) + ρ(p2 , p3 ) · ρ(p4 , p1 ). Then we say that σ satisfies the theorem of Ptolemaeus. The prototype of course is the Euclidean plane case, which was proved by the ancient Greek mathematician Claudius Ptolemaeus (Ptolemy) of Alexandria almost 1800 years ago: the inequality holds for any four points of the Euclidean plane and Ptolemaean circles are Euclidean circles. From the times of antiquity it was realized that even in the simple Euclidean case, the Ptolemaean inequality has an intrinsic importance of its own and various generalizations have been given by a variety of researchers since then. In particular, generalizations to much more abstract spaces have appeared in the last 70 years. Illustratively, we refer