A centre of mass on an infinite-dimensional Hilbert sphere need not exist, but for finite samples it always exists and can be computed in a subspace of dimension at most the sample size.
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On the existence and non-existence of centres of mass on Hilbert spheres
A centre of mass on an infinite-dimensional Hilbert sphere need not exist, but for finite samples it always exists and can be computed in a subspace of dimension at most the sample size.