Quasi-isometric modification of Gromov-Hausdorff distance

We define a distance analogous to the Gromov-Hausdorff distance that enables the comparison of arbitrary quasi-isometric spaces. We also investigate properties preserved under limits with respect to this distance, as well as properties of the entire class of metric spaces equipped with this distance. For this purpose, we introduce the notion of quasi-isometric distortion for correspondences. Using this notion, we prove that the class of all metric spaces is path-connected; in fact, any two metric spaces can be connected by a curve of finite length.

💡 Research Summary

The paper introduces a new symmetric “quasi‑isometric distance” ˆd that extends the classical Gromov–Hausdorff distance (d_GH) to arbitrary metric spaces, including non‑compact and non‑bounded ones. The authors begin by recalling the definition of d_GH, its formulation via correspondences and distortion, and the limitations of d_GH for non‑compact spaces (often infinite values, loss of coarse structure under pointed convergence).

To overcome these issues they adopt the (A,B,C)‑quasi‑isometry framework: a map f:X→Y is an (A,B)‑quasi‑isometric embedding if distances are distorted by at most a multiplicative factor A and an additive error B; two such embeddings f and g are C‑close if sup_x d_Y(f(x),g(x))<C. Spaces X and Y are (A,B,C)‑quasi‑isometric if there exist embeddings f:X→Y and g:Y→X whose compositions are C‑close to the respective identities. Using this, they define quasi‑isometric convergence X_k →_q Y: after some index N, each X_k and Y are (A_k,B_k,C_k)‑quasi‑isometric with A_k→1, B_k→0, C_k→0.

The central definition is ˆd(X,Y)=inf{r>0 | X and Y are (1+r,r,r)‑quasi‑isometric}. This function is symmetric, vanishes on identical spaces, but does not satisfy the triangle inequality in general (hence it is not a metric). Proposition 3.3 gives composition formulas for quasi‑isometry constants, and Corollary 3.4 shows a weak triangle bound ˆd(X,Z)≤2(r+r′+rr′). Crucially, Corollary 3.6 proves ˆd≤4·d_GH, so Gromov–Hausdorff convergence implies quasi‑isometric convergence, while the converse fails (e.g., ℝ versus a polygonal chain with ever‑increasing segment lengths has finite ˆd but infinite d_GH).

The authors then investigate which geometric and analytic properties are preserved under ˆd‑convergence. If X_k →_q Y, then many natural attributes of the X_k’s—total boundedness, boundedness with convergence of diameters, separability, properness (when Y is complete), intrinsic metric, geodesicity, δ‑hyperbolicity, CAT(κ) condition, etc.—are inherited by the limit Y (Corollary 3.8). This mirrors known preservation results for d_GH but now applies to the broader quasi‑isometric setting.

Since ˆd is not a genuine metric, the paper constructs an auxiliary metric D on the set M of all metric spaces (modulo the equivalence ˆd=0). Proposition 5.3 defines D, and Proposition 5.4 establishes quantitative equivalence:  if ˆd(X,Y)=r then D(X,Y)≤ln(1+2r),  if D(X,Y)=r then ˆd(X,Y)≤e^{2r}−e^{r}. Thus D induces exactly the same topology and coarse structure as ˆd, providing a metrizable framework for analysis.

The most striking structural result is Theorem 5.5. Given a correspondence R between X and Y with quasi‑isometric distortion ≤r, one can continuously deform R into a family R_t (t∈