In reading Greg Kuperberg's partial answer to this question Convex hull in CAT(0) , I started wondering if the center of gravity is always contained in the closed convex hull.

More precisely, given $n+1$ points $x_0,\dots,x_n$ in a CAT(0) space $X$, there is a center-of-gravity map $c\colon\Delta_n \to X$ (where $\Delta_n$ is the standard $n$-simplex in $\mathbb{R}^{n+1}$), defined by $$ c(a_0,a_1,\dots,a_n) = \text{the point $x \in X$ minimizing }\sum_{i=0}^n a_i d(x,x_i)^2. $$ (In a CAT(0) space, the function being minimized is strictly convex, so the center of gravity is unique.)

Is $c(a_0,a_1,\dots,a_n)$ always contained in the closed convex hull of $x_0,\dots,x_n$?

1more comments