Let $(X,d)$ be a separable complete [geodesic metric space](https://topospaces.subwiki.org/wiki/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:*
$$
\operatorname{argmin}\_{k \in K} d^2(x,k)?
$$



Existence of continuous selection for metric projection
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.


