Condition for equality of "distance to the boundary" and "distance to the set"

Question

Let $(X,d)$ be a metric space, $A \subseteq X$ and $x \notin A$. What are the condition which make true the statement

$d(x,A) = d(x,\partial A)$

If A is compact (not empty) I can think of a way to find a point $y \in A$ such that $d(x,y)=d(x,A)$. And if $X$ is a normed space that should be enough to prove $y \in \partial A$. Is there something beside this?