I want to find subgradient of the function $f({\bf x}) = {\rm inf}_{{\bf y} \in \mathcal{C}} \|{\bf x} - {\bf y}\|_2$, where $\mathcal{C}$ is a closed convex set. Now, $f$ just measures the perpendicular distance from a point to its projection on the convex compact set $\mathcal{C}$.
I am not sure how a subgradient of the given function is to be calculated. Any help is appreciated.