  Let $M$ be a simply-connected compact Riemannian symmetric space, and $U$ a
twice-differentiable function on $M$, with unique global minimum at $x^* \in
M$. The idea of the present work is to replace the problem of searching for the
global minimum of $U$, by the problem of finding the Riemannian barycentre of
the Gibbs distribution $P_{\scriptscriptstyle{T}} \propto \exp(-U/T)$. In other
words, instead of minimising the function $U$ itself, to minimise
$\mathcal{E}_{\scriptscriptstyle{T}}(x) = \frac{1}{2}\int d^{\scriptscriptstyle
2}(x,z)P_{\scriptscriptstyle{T}}(dz)$, where $d(\cdot,\cdot)$ denotes
Riemannian distance. The following original result is proved : if $U$ is
invariant by geodesic symmetry about $x^*$, then for each $\delta < \frac{1}{2}
r_{\scriptscriptstyle cx}$ ($r_{\scriptscriptstyle cx}$ the convexity radius of
$M$), there exists $T_{\scriptscriptstyle \delta}$ such that $T \leq
T_{\scriptscriptstyle \delta}$ implies $\mathcal{E}_{\scriptscriptstyle{T}}$ is
strongly convex on the geodesic ball $B(x^*,\delta)\,$, and $x^*$ is the unique
global minimum of $\mathcal{E}_{\scriptscriptstyle{T\,}}$. Moreover, this
$T_{\scriptscriptstyle \delta}$ can be computed explicitly. This result gives
rise to a general algorithm for black-box optimisation, which is briefly
described, and will be further explored in future work.
