A New One-Point Metric on Ptolemaic Spaces

In this paper, given \(p, p_1, p_2, \ldots , p_k\) $p,p_1,p_2,\dots ,p_k$ in a Ptolemaic space (X, d), we introduce a new one-point metric \(\begin{aligned} \tilde{S}_p(x,y)=\log \left( 1+\frac{d(x,y)}{\sqrt{1+d(x,p)}\sqrt{1+d(y,p)}}\right) \end{aligned}\) $$\begin{array}{c}\hfill {\tilde{S}}_p(x,y)=log\left(1,+,\frac{d(x,y)}{\sqrt{1+d(x,p)}\sqrt{1+d(y,p)}}\right)\end{array}$$ and the average of such kind of metrics \(\begin{aligned} \tilde{S}(x,y)=\frac{1}{k}\sum _{i=1}^k \tilde{S}_{p_i}(x,y) \end{aligned}\) $$\begin{array}{c}\hfill \tilde{S}(x,y)=\frac{1}{k}\sum _{i=1}^k{\tilde{S}}_{p_i}(x,y)\end{array}$$ for \(x,y\in X\) $x,y\in X$ . We prove the Gromov hyperbolicity of these metrics. We compare the metric \(\tilde{S}_p\) ${\tilde{S}}_p$ with the metric \(S_p\) $S_p$ and the metric d of the base metric space (X, d). We show the quasiconformality of the identity map \(\mathrm{{id}}:(X,d)\rightarrow (X,\tilde{S}_p)\) $\mathrm{id}:(X,d)\to (X,{\tilde{S}}_p)$ and discuss the relations between the identity map and bilipschitzian maps, quasisymmetric maps and quasi-Möbius maps.