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May 28 at 7:43 vote accept Nate River
May 28 at 7:43 comment added Nate River Ah uniqueness would do it…
May 28 at 5:23 comment added Guido De Philippis It follows from uniqueness, if $x_k\to x$ and $\pi(x_k)\to y$, then it is easy to see that $y$ realizes the distance and so it has to coincide with $\pi(x)$.
May 28 at 5:20 history edited Guido De Philippis CC BY-SA 4.0
Corrected the formula for the gradient
May 27 at 20:46 comment added Nate River Thanks for the answer! Is it obvious that $x - \pi(x)$ is continuous at any point of differentiability? Having trouble showing this formally.
May 27 at 20:44 vote accept Nate River
May 27 at 20:45
May 27 at 16:43 history answered Guido De Philippis CC BY-SA 4.0