The ptolemaic characteristic of tetrads and quasiregular mappings

We consider the Ptolemaic characteristic of quadruples of disjoint nonempty compact subsets (generalized tetrads). The main theorem of this article asserts that an arbitrary multivalued mapping $F$ from ${\Bbb R}^n$ onto itself such that the images of distinct points are disjoint and each of them contains at most two distinct points is the inverse of a $K$-quasimeromorphic mapping if and only if $F$ admits a controllable upper bound for the distortion of the Ptolemaic characteristic of tetrads.