The Gromov–Hausdorff distance between spheres

Abstract

We provide general upper and lower bounds for the Gromov–Hausdorff distance dGH(Sm, Sn) between spheres Sm and Sn (endowed with the round metric) for 0 < m < n < 1. Some of these lower bounds are based on certain topological ideas related to the Borsuk–Ulam theorem. Via explicit constructions of (optimal) correspondences, we prove that our lower bounds are tight in the cases of dGH(S0, Sn), dGH(Sm, S∞), dGH(S1, S2), dGH.S1; S3) and dGH(S2; S3). We also formulate a number of open questions.