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 $\mathbb{R}^3$). 



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> 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 $\mathbb{R}^3$?
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A reference would be very helpful.



Pogorelov's rigidity theorem vs Cohn-Vossen rigidity theorem
Below is the answer from the comments. First some terminology. 


A *convex surface* is the boundary of a compact convex body in $\mathbb R^3$. 
Each convex surface comes with two metrics: the path-metric and the metric obtained by restricting the distance function on $\mathbb R^3$, which we call *intrinsic* and *extrinsic*, respectively. A homeomorphism $f:(A, d\_A)\to (B, d\_B)$ of metric spaces is a *$\delta$-isometry*
if $$|d\_B(f(x), f(y))-d\_A(x, y)| < \delta$$ for any $x, y\in A$. Of course, $f$ is an isometry if it a $\delta$ isometry for every $\delta$.


A stability theorem of Volkov (translated to English as an appendix in 
[A. D. Alexandrov selected works. Part II, Chapman & Hall/CRC,
Boca Raton, FL, 2006, Intrinsic geometry of convex surfaces]
says that if $S\_1, S\_2$ are convex surfaces with intrinsic metrics $\rho\_1$, $\rho\_2$, and extrinsic metrics
$d\_1, d\_2$ , and if $f:(S\_1 , \rho\_1) \to (S\_2, \rho\_2)$ is an $\epsilon$-isometry, then $f:(S\_1 , d\_1) \to (S\_2, d\_2)$ is an $C\_1 \epsilon^\beta$-isometry where $C$ depends onto on diameters of $\rho\_1$, $\rho\_2$ and $\beta$ is a positive universal constant.


By Theorem 2 in [Alestalo, P.; Trotsenko, D. A.; Väisälä, J. Isometric approximation. Israel J. Math. 125 (2001), 61–82] any $\delta$-isometry between compacta in $\mathbb R^n$ can be approximated
by the restriction of an isometry of $\mathbb R^n$ with the additive error at most $C\_2\sqrt{\delta}$, where $C\_2$ depends only on $n$ and the diameters of the compacta. Here the metric on a compactum is the restriction of the distance on $\mathbb R^n$.


Thus every path-isometry of convex surfaces extends to an isometry of $\mathbb R^3$.


