The coefficient of quasimöbiusness in Ptolemaic spaces

In ptolemaic spaces the class of $\eta$-quasimöbius mappings $f: X\to Y$ with control function $\eta(t)= C \max\{ t^{\alpha}, t^{1/\alpha}\}$ may be completely characterized by the inequality $ K^{-1}\leq (1 + \log P(fT))/(1+ \log P(T)) \leq K$ for all tetrads $T\subset X$ where $P(T)$ denotes the ptolemaic characteristic of a tetrad. The number $K$ has properties quite similar to those of coefficients of quasiconformality, so the concept of $K$-quasimöbius mapping may be introduced. In particular, the stability theorem is proved for $(1+\varepsilon)$-quasimöbius mappings in $\bar{R}^n$.