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arXiv · 2607.13667

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

Shahin Tavakoli
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Abstract

Fr\'echet 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.

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Shahin Tavakoli. 2026-07-15. On the existence and non-existence of centres of mass on Hilbert spheres. https://arxiv.org/abs/2607.13667

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