A criterion for the unique determination of domains in Euclidean spaces by the metrics of their boundaries induced by the intrinsic metrics of the domains

We say that a domain $U\subset\mathbb R^n$ is uniquely determined by the relative metric (which is the extension by continuity of the intrinsic metric of the domain on its boundary) of its Hausdorff boundary if any domain $V\subset\mathbb R^n$ such that its Hausdorff boundary is isometric in the relative metric to the Hausdorff boundary of $U$, is isometric to $U$ in the Euclidean metric. In this paper, we obtain the necessary and sufficient conditions for the uniqueness of determination of a domain by the relative metric of its Hausdorff boundary.