In reading Greg Kuperberg's partial answer to this question [Convex hull in CAT(0)](https://mathoverflow.net/questions/6627/convex-hull-in-cat0) ,
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 $x\_0,\dots,x\_n$?



Is the center of gravity in a CAT(0) space contained in the convex hull?
This is expanding on @user35593 's comment above.


Let $\bar{C}$ be the closure of the convex hull of $x\_0,\dots,x\_n$, and let $x'$ be the projection of $x$ onto $\bar{C}$. By Bridson and Haefliger, Proposition II.2.4 (which is easy), projection onto the convex hull is a retraction that does not increase distances. In particular, $d(x',x\_i) \le d(x,x\_i)$. Since $x$ was the unique minimizer of a positive combination of the distances, we must have $x=x'$.


