# Existence of continuous selection for metric projection

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: $$\operatorname{argmin}_{k \in K} d^2(x,k)?$$

• This rules out many natural conditions -- will you give an example of a condition of the form that you're looking for? Jan 23 '21 at 14:11
• Unfortunately that's the trouble, I'm only familiar with the conditions I ruled out or those for Micheal's Selection theorem? Jan 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.