Generalization to length spaces and required density assumptions

Investigate whether the lower bounds and equality results between the Gromov–Hausdorff distance and the Hausdorff distance proved for compact, connected metric graphs extend to general length metric spaces, and ascertain the precise density assumption (expressed as a threshold on the Hausdorff distance between the space and the subset) under which such analogues hold; in particular, determine this for manifolds with boundary.

Background

The paper proves graph-specific results: under suitable boundary-density conditions, d_GH(G,X) ≥ min{ d_H(G,X), (1/12)·e(G) }, implying equality when d_GH(G,X) is below the threshold. The question is whether similar phenomena occur in broader geodesic (length) spaces.

A natural target for generalization mentioned by the authors is manifolds with boundary, where identifying appropriate geometric parameters and density thresholds would extend the applicability of their methods beyond graphs.