On the existence and non-existence of centres of mass on Hilbert spheres

Fréchet means and $L^p$ centres of mass provide notions of average location in metric spaces. On finite-dimensional spheres, existence follows from compactness.
On infinite-dimensional spheres, it is not known whether a centre of mass always exists. We show that this is not always the case, and give a simple assumption under which a centre of mass exists. We then show that finding the sample centre of mass of data $x_1, \ldots, x_n$ on the sphere is always an optimisation problem on a subsphere of manifold dimension at most $n$, regardless of the potentially infinite dimension of the sphere. We conclude with some statistical implications.