The well known Pogorelov’s rigidity theorem says that if two convex closed surfaces in Euclidean 3-space are isometric to each other being equipped with their intrinsic metrics, then they are congruent, namely there is an isometry of the Euclidean space which maps one surface onto the other.
Are there counter examples to Pogorelov’s theorem in higher dimensions?
There is definitely no counterexample for convex polyhedra and no counterexample for bodies with smooth boundary with non-degenerate second fundamental form. (In the latter case even small open subsets of the boundary cannot be deformed, the argument can be found e.g. in Spivak volume 5.)
I do not know if the question was asked somewhere in full generality, although it sounds very natural.
If one could prove a sort of stability estimate: if the intrinsic metrics of surfaces are $\epsilon$-close, then the bodies are $\delta$-close, where $\delta$ depends on $\epsilon$ and some "rough" geometric characteristics of the bodies like their diameter, then one would get the uniqueness in the general case by approximation. Such a stability estimate is claimed to be proved in Volkov's article reproduced as appendix to the new edition of Alexandrov's "Convex polyhedra", but the proof is flawed.
