The Möbius midpoint condition as a test for quasiconformality and the quasimöbius property

The Möbius midpoint condition, introduced by Goldberg in 1974 as a criterion for the quasisymmetry of a mapping of the line onto itself and considered by Aseev and Kuzin in 1998 in the same role for the topological embeddings of the line into $\mathbb R^n$, yields no information on the quasiconformality or quasisymmetry of a topological embedding of $\mathbb R^k$ into $\mathbb R^n$ for $1<k\le n$. In this article we introduce a Möbius-invariant modification of the midpoint condition, which we call the “Möbius midpoint condition” $\mathrm{MMC}(f)\le H<1$. We prove that if this condition is fulfilled then every homeomorphism of domains in $\overline{\mathbb R^n}$ is $K(H)$-quasiconformal, while a topological embedding of the sphere $\overline{\mathbb R^k}$ into $\overline{\mathbb R^n}$ (for $1\le k\le n$) is $\omega_H$-quasimöbius. The quasiconformality coefficient of $K(H)$ and the distortion function $\omega_H$ depend only on $H$ and are expressed by explicit formulas showing that $K(H)\to1$ and $\omega_H\to\mathrm{id}$ as $H\to1/2$. Since $\mathrm{MMC}(f)=1/2$ is equivalent to the Möbius property of $f$, the resulting formulas yield the closeness of the mapping to a Möbius mapping for $H$ near $1/2$.