Weak triangle inequality for the graph-embedding d_r family

Determine whether the family of functions {d_r}_{r>0} defined on the space of embedded graphs Φ(ℝ^N) (the space of graphs in ℝ^N as in Galatius, MR2784914)—where for r>0 and G,H∈Φ(ℝ^N), P_r(G,H) consists of maps φ:U→φ(U) with U⊂H and φ(U)⊂G open, H∩B_r⊂U, G∩B_r⊂φ(U), and d_r(G,H)=inf_{φ∈P_r(G,H)} sup_{p∈(H∩B_r)∪φ^{-1}(G∩B_r)} |p−φ(p)| (with d_r=∞ if P_r(G,H)=∅)—satisfies the weak triangle inequality: for all r_1,r_2,r_3>0 and G_1,G_2,G_3∈Φ(ℝ^N), if d_{r_1+r_2}(G_1,G_2)<r_3 and d_{r_1+r_2+r_3}(G_2,G_3)<r_2, then d_{r_1}(G_1,G_3) ≤ d_{r_1+r_2}(G_1,G_2) + d_{r_1+r_2+r_3}(G_2,G_3).

Background

To develop a quasi-uniformity from families {d_r}_{r>0}, the paper introduces a weaker triangle inequality that suffices for key constructions. In the case of the space of graph embeddings Φ(ℝ^N) (as defined by Galatius, MR2784914), the authors define a natural family {d_r} using Galatius-type triples and prove that this family satisfies only the weaker triangle inequality, which already recovers the C^0-topology on Φ(ℝ^N).

However, the stronger ‘weak triangle inequality’ used earlier in the framework (and satisfied in other examples such as spaces of submanifolds) is not verified for Φ(ℝ^N). Establishing it would align the graph-embedding example with the full set of axioms introduced for {d_r}_{r>0}, potentially strengthening uniformity and topology correspondences in this setting.