Given a [CAT(0) space](http://en.wikipedia.org/wiki/CAT%2528k%2529_space) $X$ 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](http://en.wikipedia.org/wiki/Hausdorff_distance) between $A\_1$ and $A\_2$?



Stability of midpoints in CAT(0) spaces
No, even if $X=\mathbb R^2$.


Let $A\_1$ be (the convex hull of) 4 points with coordinates $(\pm 1,\pm 1)$. Then $m(A\_1)=(0,0)$, as the 4 points are on the circle $S\_1$ of radius $\sqrt 2$ centered at $(0,0)$. Shift $S\_1$ a small distance $\varepsilon$ in the horizontal direction, denote the resulting circle by $S\_2$. For each vertex of $A\_1$, mark its nearest point on $S\_2$. The marked points are vertices of a convex quadrangle $A\_2$ inscribed in $S\_2$ and containing its center $(\varepsilon,0)$. Hence $m(A\_2)=(\varepsilon,0)$ but the Hausdorff distance between $A\_1$ and $A\_2$ is $\approx\varepsilon/\sqrt 2$.


