Separation theorems of Teichmüller type

A ring domain (annulus) in the complex plane contains a round (genuine) subring of the form $r_1<|z-a|<r_2$ if the ring domain has a large enough modulus $m.$ Moreover, the subring can be taken so that $\log(r_2/r_1) \ge m-C,$ where $C$ is an absolute constant. This sort of result was first proved by Teichmüller. In [1], we introduced a notion of semi-annulus and its modulus and applied it to study boundary continuity of homeomorphisms of a disk or a half-plane.
In the present talk, we extend these result into the $n$-dimensional case. Indeed, we have similar results for rings and semi-rings in $\mathbb{R}^n.$ This is joint work with Anatoly Golberg.