Let $(X,d)$ be a separable complete geodesic metric space and let $K$ be a compact (nonempty) 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: $$ \operatorname{argmin}_{k \in K} d^2(x,k)? $$

1$\begingroup$ 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? $\endgroup$– Catologist_who_flies_on_MondayJan 23 '21 at 14:16
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.