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Let $(X,d)$ be a separable complete geodesic metric space and let $K$ be a compact (non-empty) subset of $X$. Without assuming things like linearity, the convexity of $K$, and locally convexity, under what conditions can we guarantee that there is a continuous selection for the metric projection problem: $$ \operatorname{argmin}_{k \in K} d^2(x,k)? $$

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    $\begingroup$ This rules out many natural conditions -- will you give an example of a condition of the form that you're looking for? $\endgroup$
    – Matt F.
    Jan 23 '21 at 14:11
  • $\begingroup$ Unfortunately that's the trouble, I'm only familiar with the conditions I ruled out or those for Micheal's Selection theorem? $\endgroup$ Jan 23 '21 at 14:16
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In general, $argmin$ is not continuous. Even on the real line, if I take $K$ to be two distinct points, say $K=\{-1,1\}$, then $argmin_{k\in K} d^2(x,k)$ is not continuous. This is why convexity is so important. Without assuming convexity, little is guaranteed in this case.

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