Given a CAT(0) 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 between $A_1$ and $A_2$?

Given a CAT(0) 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 between $A_1$ and $A_2$?

Given a CAT(0) 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 $d(m(A_1),m(A_2))$ smaller than the Hausdorff distance between $A_1$ and $A_2$?

Given a CAT(0) 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 between $A_1$ and $A_2$?