TITLE: Is the center of gravity in a CAT(0) space contained in the convex hull?
QUESTION [10 upvotes]: 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 $x_0,\dots,x_n$?

REPLY [11 votes]: 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 $x=x'$.