Given a global NPC (no positive curvature) metric space $(S,d)$, and $x_1,\ldots,x_n\in S$, let $\bar x_n$ be their barycentre, i.e., the unique minimized of $\sum_{i=1}^n d(x,x_i)^2, x\in S$, and let $m_1,m_2, \ldots,m_n$ be the sequence of iterated means defined as follows: $m_1=x_1$ and, for $k=2,\ldots,n$, $m_k$ is the unique minimiser of $\frac{k}{k+1}d(x,m_{k-1})^2+\frac{1}{k+1}d(x,x_k)^2, x\in S$.
If $(S,d)$ was a Euclidean space, then $\bar x_n$ and $m_n$ would coincide, but in general they do not (and the value of $m_n$ depends on the pre-ordering of $x_1,\ldots,x_n$. Is it possible, though, to show an inequality of the form: $$d(\bar x_n,m_n)\leq CD_n/n,$$ where $D_n$ is the diameter of the set $\{x_1,\ldots,x_n\}$ And $C$ is a positive constant that may only depend on some characteristics of the space $(S,d)$? Thanks!