Non unique vector for Infimum distance

Question

Let a function $d: \mathbb{R}^n \rightarrow \mathbb{R}$ be define as $x \mapsto inf_{a \in A} \vert\vert x-a \vert\vert$ where $a \in A \subset \mathbb{R}^n$ and $A$ is closed. Then I know that there is some vector $a_x$ for every $x$ such that $inf_{a \in A} \vert\vert x-a \vert\vert = \vert\vert x-a_x \vert\vert$. But is this vector $a_x$ unique in general?

Answer 1

Certainly not unique in general.

You may think of the following example : $n=2$, $A$ is the unit circle (which is indeed close in $\mathbb{R}^2$) and $x=0$.