The Ptolemaic Characteristic of Tetrads and Quasiregular Mappings

We consider the Ptolemaic characteristic ofquadruples of disjoint nonempty compact subsets(generalized tetrads).The main theorem of this article asserts thatan arbitrary multivalued mapping  $ F $ from  $ {𝕉}^{n} $ onto itself such thatthe images of distinct points are disjointand each of them contains at most two distinct pointsis the inverse of a  $ K $ -quasimeromorphic mappingif and only if  $ F $ admits a controllable upper bound for the distortionof the Ptolemaic characteristic of tetrads.