Hilbert-space realizability of atomless laws

Determine which atomless probability measures on [0,∞) can be realized as distance distributions of independent points in a Hilbert space.

Background

The main theorem concerns realizations in arbitrary complete separable metric spaces, while the original distance-distribution question arose in the more restrictive setting of Hilbert spaces. The paper leaves open the characterization of atomless laws that can occur as distances between two independent points sampled from a probability measure on a Hilbert space.

References

Which atomless laws can be realised by distances in a Hilbert space? This is the setting of the original question in .

— Realising atomless laws as distance distributions on metric measure spaces  (2608.28330 - Thäle et al., 28 Aug 2026) in Section 5, item (iii) of the enumerated future-research questions