TITLE: What are the applications of the Mazur-Ulam Theorem?
QUESTION [14 upvotes]: Every bijective isometry between normed spaces is affine. This well-known and beautiful statement, the Mazur-Ulam Theorem, was proved in 1932, but the proof has been simplified and polished in years, by contributions of more people (Lyusternik,  Borsuk, Vogt, Väisälä, and others I'm not aware of) until the final form of a little gem (the last version I know, by Bogdan Nica, can be explained in conversation).
Reaching a linearity property from a metric assumption is something peculiar. Another  result of this "from metric to linear" form, that comes to my mind, is the linearity of the metric projector on a subspace of a Hilbert space. However, the latter is a fundamental result with very important and rich consequences on the structure of Hilbert spaces, and applications in the Calculus of Variations. On the opposite, I know no application of Mazur-Ulam theorem, and I would be really glad to learn some. Applications of any generalization of it are also welcome.
edit.  Of course, knowing that surjective isometries on a normed space are linear has a clear foundational application, since any theory about isometries of normed spaces can assume linearity, and exploit the linear theory of operators. Then the question is: what natural sources of (a priori non linear) isometries of normed spaces are there?

REPLY [6 votes]: The Mazur-Ulam theorem is used in  Bader, Uri; Furman, Alex; Gelander, Tsachik; Monod, Nicolas Property (T) and rigidity for actions on Banach spaces. Acta Math. 198 (2007), no. 1, 57–105 in order to simplify the theory of isometric actions on Banach spaces. In this connection it is worth mentioning that the question on an analogue of the Mazur-Ulam theorem for convex sets (which they ask on page 88) was answered by Schechtman, see Generalizing the Mazur-Ulam theorem to convex sets with empty interior in Banach spaces