On asymptotic dimension and a property of Nagata

In this note we prove that every metric space $(X,d)$ of asymptotic dimmension at most $n$ is coarsely equivalent to a metric space $(Y,D)$ that satisfies the following property of Nagata: For every $n+2$ points $y_1,\mathrm{.}\mathrm{.}\mathrm{.},y_{n+2}$ in $Y$ and for every $x$ in $Y$ there exist two different $i,j$ such that $D(y_i,y_j)\le D(x,y_i)$. This solves problem 1400 of the book Open problems in Topology II.