A new one-point metric on Ptolemaic spaces (Q6561364)

A metric space \((X,d)\) is called \textit{Ptolemaic} if it satisfies the Ptolemy inequality that relates the six distances determined by any four points. Motivated by the previous work of \textit{M. Bonk} and \textit{B. Kleiener} [J. Differ. Geom. 61, No. 1, 81--106 (2002; Zbl 1044.37015)] and \textit{Z. Zhang} and \textit{Y. Xiao} [J. Math. Anal. Appl. 478, No. 2, 445--457 (2019; Zbl 1425.30045)] the authors construct a new one-point metric \(\tilde{S}_p(x,y)\) on a Ptolemaic space \(X\) and its \(k\)-point average \(\tilde{S}(x,y)\). They prove that the metric space \((X, \tilde{S}_p)\) is Gromov hyperbolic with the hyperbolicity constant \(\delta = \log 3\) (see Theorem 2) and that the space \((X, \tilde{S})\) is Gromov hyperbolic with \(\delta = 3 \log 2 + \log 3\) (in Theorem 3). They also show \(1\)-quasiconformality of the identity map \((X, d) \to (X, \tilde{S}_p)\) (see Theorem 5) and discuss the relations between the identity map and bi-Lipschitz maps, quasisymmetric maps and quasi-Möbius maps.