On the unique determination of domains by the condition of the local isometry of the boundaries in the relative metrics. II

We prove the theorem on the unique determination of a strictly convex domain in $\mathbb R^n$, where $n \ge 2$, in the class of all $n$-dimensional domains by the condition of the local isometry of the Hausdorff boundaries in the relative metrics, which is a generalization of A. D. Aleksandrov's theorem on the unique determination of a strictly convex domain by the condition of the (global) isometry of the boundaries in the relative metrics.
We also prove that, in the cases of a plane domain $U$ with nonsmooth boundary and of a three-dimensional domain $A$ with smooth boundary, the convexity of the domain is no longer necessary for its unique determination by the condition of the local isometry of the boundaries in the relative metrics.