Given a [CAT(0) space][1] $X$ and a compact, convex subset $A$ of $X$. One can define its midpoint $m(A)$ as the point, at which the following function attains its minimum.

$f:A\rightarrow \mathbb{R}\qquad x\mapsto \sup\{d(x,y)|y\in A\}$

One can show, that there is a unique such point. So my question is:

Given two compact,convex subset $A_1,A_2\subset X$. Is $d(m(A_1),m(A_2))$ less or equal to the [Hausdorff distance][2] between $A_1$ and $A_2$?


  [1]: http://en.wikipedia.org/wiki/CAT%2528k%2529_space
  [2]: http://en.wikipedia.org/wiki/Hausdorff_distance