A New One-Point Metric on Ptolemaic Spaces

In this paper, given p, p(1), p(2),...,p(k) in a Ptolemaic space (X, d),we introduce a new one-point metric S-p(x, y) = log (1 + d(x, y)/root 1 + d (x, p)root 1 + d(y, p)) and the average of such kind of metrics S(x, y) = 1/k & sum;(k )(i=1) S-pi(x, y) for x, y is an element of X. We prove the Gromov hyperbolicity of these metrics. We compare the metric S-p with the metric S-p and the metric d of the base metric space (X, d). We show the quasiconformality of the identity map id : (X, d) -> (X, S-p) and discuss the relations between the identity map and bilipschitzian maps, quasisymmetric maps and quasi-Mobius maps.