There is the following rigidity theorem of Cohn-Vossen as stated on p. 86 of these lecture notes: http://www.math.brown.edu/~deigen/chern.pdf

**Any isometry between two closed smooth convex surfaces (equipped with the induced path metrics) in the Euclidean space $\mathbb{R}^3$ is established by an isometry of $\mathbb{R}^3$.**

In his book "Extrinsic geometry of covex sufaces" Pogorelov generalizes (partly?) this result to a pair of convex surfaces without any extra assumptions on regularity in the following form:

**If two closed convex surfaces in $\mathbb{R}^3$ are isometric with respect to the induced inner (path) metric then they are congruent, i.e. there is an isometry of $\mathbb{R}^3$ which maps one surface to another one.**

It seems to me that the second statement, being specialized to smooth surfaces, is weaker than the first one.

ADDED: Consider say the special case when the two surfaces are smooth and coincide with each other. Then the Pogorelov theorem is trivial in this case, while the Cohn-Vossen theorem gives non-trivial information on isometries of the surface (all of them extend to isometries of $\mathbb{R}^3$).

Thus I am wondering whether any isometry between two closed (not necessarily smooth) convex surfaces in the Euclidean space $\mathbb{R}^3$ is established by an isometry of $\mathbb{R}^3$. In particular, is it true that any self-isometry of a closed convex surface is established by an isometry of $\mathbb{R}^3$?

A reference would be very helpful.