Let $(X,d)$ be a metric space. A *barycenter* of a Borel probability measure $\mu$ on $X$ is a minimizer of the function
\begin{equation}
\begin{split}
f \colon X & \to \mathbb{R}\\
x &\mapsto \frac 12 \int_X d(x,\cdot)^2 \mathop{}\!\mathrm{d}\mu~.
\end{split}
\end{equation}
NB: A barycenter is also sometimes called a *center of mass*, a *Fréchet mean* or a *Karcher mean*.

In general, neither existence nor uniqueness of barycenters hold.

I am interested in the case where $X$ is a **possibly incomplete** simply connected Riemannian manifold (without boundary) of negative sectional curvature, and moreover is uniquely geodesic (any two points are connected by a unique minimizing geodesic). It is not hard to show that uniqueness of barycenters holds, because $f$ is a strictly convex function. I wonder if existence holds, especially in the case where $\mu$ is finitely supported (I would like to define the barycenter of any finite set of points).

**Question:** Does existence hold in this case?

Here are a couple more observations. Note that I could have merely assumed $X$ to be a metric space -- a (possibly incomplete) uniquely geodesic (locally compact) $\mathrm{CAT}(0)$ metric space, but maybe the techniques of Riemannian geometry can be useful. In the setting of metric spaces, the papers that I have looked at always assume completeness. Karcher's original proof of existence in the setting of Riemannian manifolds also assumes completeness, but maybe it can be adapted. It consists in observing that if the support of $\mu$ is contained in some closed convex ball $B = \overline{B(x_0,r)}$, then the gradient of $f$ which is given by \begin{equation} \mathrm{grad} f (x) = -\int_X \exp_x^{-1}(\cdot) \mathop{}\!\mathrm{d}\mu \end{equation} is pointing outwards on the boundary of $B$, therefore a minimum of $f$ on $B$ must be attained in the interior. In my situation, the problem is that a closed ball is not necessarily compact. I thought I could adapt this argument by replacing a convex ball by a compact convex set, but I am unsure whether it is true that any finite set of points is contained in some compact convex set in this class of manifolds...

**Edit:** Misha pointed out to me in the comments below that Teichmüller space with the Weil-Petersson metric is exhausted by compact convex sets. S. Wolpert proves it in "Geodesic length functions and the Nielsen problem". This solves my problem in this particular situation (according to the discussion above). I wonder that a uniquely geodesic simply connected Riemannian manifold of negative sectional curvature is always exhausted by compact convex sets.