Means in complete manifolds: uniqueness and approximation

Let $M$ be a complete Riemannian manifold, $N\in \text{NN}$ and $p\ge 1$. We prove that almost everywhere on $x=(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_N)\in M^N$ for Lebesgue measure in $M^N$, the measure $\text{di}\mu (x)=\text{f}1N{\sum }_{k=1}^N{\text{d}}_{x_k}$ has a unique $p$-mean $e_p(x)$. As a consequence, if $X=(X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_N)$ is a $M^N$-valued random variable with absolutely continuous law, then almost surely $\mu (X(\text{om}))$ has a unique $p$-mean. In particular if $(X_n{)}_{n\ge 1}$ is an independent sample of an absolutely continuous law in $M$, then the process $e_{p,n}(\text{om})=e_p(X_1(\text{om}),\mathrm{.}\mathrm{.}\mathrm{.},X_n(\text{om}))$ is well-defined. Assume $M$ is compact and consider a probability measure $\nu$ in $M$. Using partial simulated annealing, we define a continuous semimartingale which converges to the set of minimizers of the integral of distance at power $p$ with respect to $\nu$. When the set is a singleton, it converges to the $p$-mean.