Characterization of finite sampled distance-matrix laws

Characterize, for each fixed integer n≥3, the probability measures on [0,∞)^{\binom{n}{2}} that can occur as the joint law of the pairwise distances among n independent points with a common distribution on a metric space.

Background

The paper studies only the two-point distance distribution. A broader unresolved inverse problem concerns the joint distribution of all pairwise distances in a finite independent sample. Any admissible law must satisfy the triangle inequalities, be invariant under relabeling of the sampled points, and possess compatible extensions to larger sample sizes. The case n=3 had previously been proposed as a natural next problem, but no general characterization is supplied here.

References

For a fixed n\geq3, which probability measures on {\binom{n}{2}$ can occur as the joint law of \bigl(d(X_i,X_j)\bigr)_{1\leq i<j\leq n} for independent points X_1,\ldots,X_n with common law? Such laws must satisfy the triangle inequalities, be invariant under relabelling the points, and admit compatible extensions to larger sample sizes.

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