These are the definition of Hausdorff distance and Hausdorff semi-distance for subsets of a metric space $X$.
Hausdorff semi-distance of two subsets $A, B \subset X$ is defined as below: $d(A \mid B) := \sup \lbrace d(x,B) : x \in A \rbrace $, where $d(x, B) := \inf \lbrace d(x,y) : y \in B \rbrace $.
Also Hausdorff distance for the subsets $A, B \subset X$ is defined as follow : $d_H (A,B) := \max\lbrace d(A\mid B), d(B \mid A) \rbrace $ .
I want to know that are there any sufficient conditions on $A$ and $B$ which imply that $d(A\mid B) = d(B \mid A)$ ?
I guess that for convex sets the equality hold, but I can't prove my intuition. Thanks in advance for any help/comment.