The Gromov-Hausdorff distance between $l^p$-products of metric spaces

This paper studies $l^p$-products of metric spaces and provides estimates for the Gromov-Hausdorff distances between them. The case of linear products is considered separately, and sufficient conditions for attainability of the estimates are given for it. Examples of calculating the Gromov-Hausdorff distance between flat tori are given. It is proved that for any metric space $X$ of density $d(X)$, the Gromov-Hausdorff distance between it and its $l^\infty$-product (in which the number of factors corresponds to $d(X)$) is equal to half its diameter.

💡 Research Summary

The paper investigates the Gromov–Hausdorff (GH) distance between ℓ^p‑products of metric spaces, providing both general two‑sided estimates and sharp results for special cases. After recalling the definition of the GH distance and its continuous analogue d_cGH, the authors introduce the ℓ^p‑product construction: given a family {X_n} of metric spaces with ∑ diam(X_n)^p < ∞ (for 1 ≤ p < ∞) they define (ℓ^p)∏X_n as the Cartesian product equipped with the ℓ^p‑norm of the component distances; for p = ∞ they use the supremum norm.

In Section 2 they prove Lemma 2.2, which shows that any collection of correspondences R_n between X_n and Y_n yields a correspondence R between the products, and the distortion of R is bounded by the ℓ^p‑norm of the individual distortions. Consequently, \