Support function of a subset $S$ of $\mathbb{R}^n$ is given by $$\delta_S (x)=\sup\{\langle x,y\rangle ∶y∈S\}.$$ If $S$ is closed, it is clear that there is some $y_1\in S$ for which this supremum attains.
Now suppose $S$ is a closed convex set. Then for any $x\in\mathbb{R}^n$ there is a unique $y_2\in S$ such that $$d(x, S)=\inf\{\Vert x-y\Vert : y\in S\}$$ is equal to $\Vert x-y_2\Vert.$
Now my question is if $S$ is closed and convex, for an arbitrary $x\in S,$ is there any relationship between $y_1$ and $y_2$?
If yes, what is it? If not, can you explain why not?
Added later: In general, $y_1$ is not unique. But I guess there can be some interesting possibilities for it. For instance, can we choose them to be "some sort of" diametrically opposite points of $S$?