Coverage in the unresolved intrinsic-dimension range for general metric spaces

Establish coverage of the centre set for reference depth in general metric spaces in the remaining intrinsic-dimension range, including the case D=2, beyond the regimes D>2(2−γ) and D<2 covered under additional local conditions.

Background

The paper studies simultaneous inference for rankings of random objects using the reference depth, which is defined from pairwise dissimilarities. In general metric spaces, the authors establish coverage of the centre set when the intrinsic dimension D satisfies D>2(2−γ), and also under further local assumptions when D<2. The discussion identifies the intermediate range, including D=2, as unresolved.

Resolving this problem would complete the coverage theory for centre sets across the full range of intrinsic dimensions considered by the metric-space framework.

References

Outside Euclidean space, the remaining range, including $D=2$, is open.

— Simultaneous Inference for Centre-Outward Rankings of Random Objects  (2610.18533 - Lyu et al., 13 Oct 2026) in Discussion, Section 7

In the location family the argument of Theorem~\ref{thm:1dlower} gives only a lower bound of constant order when $d\ge2$ (Supplementary Section~S3), so it remains open whether the logarithmic factors in Theorem~\ref{thm:highd}(i) are needed.

— Simultaneous Inference for Centre-Outward Rankings of Random Objects  (2610.18533 - Lyu et al., 13 Oct 2026) in Discussion, Section 7