  The article contains the results of the author's recent investigations of
rigidity problems of domains in Euclidean spaces carried out for developing a
new approach to the classical problem of the unique determination of bounded
closed convex surfaces [A.V.Pogorelov, Extrinsic Geometry of Convex Surfaces,
AMS, Providence (1973)], rather completely presented in [A.P.Kopylov, On the
unique determination of domains in Euclidean spaces, J. of Math. Sciences, 153,
no.6, 869-898 (2008)].
  We give a complete characterization of a plane domain $U$ with smooth
boundary (i.e., the Euclidean boundary $\mathop{\rm fr}U$ of $U$ is a
one-dimensional manifold of class $C^1$ without boundary) that is uniquely
determined in the class of domains in $\mathbb R^2$ with smooth boundaries by
the condition of the local isometry of the boundaries in the relative metrics.
If $U$ is bounded then the convexity of $U$ is a necessary and sufficient
condition for the unique determination of this kind in the class of all bounded
plane domains with smooth boundaries. If $U$ is unbounded then its unique
determination in the class of all plane domains with smooth boundaries by the
condition of the local isometry of the boundaries in the relative metrics is
equivalent to its strict convexity.
  In the last section, we consider the case of space domains.
  We prove a 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.
