The Gromov-Hausdorff distance between spheres

We provide general upper and lower bounds for the Gromov-Hausdorff distance d(GH)(S-m, S-n) between spheres S-m and S-n (endowed with the round metric) for 0 <= m < n <= infinity. 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 d(GH)(S-0, S-n), d(GH)(S-m, S-infinity), d(GH)(S-1, S-2), d(GH)(S-1, S-3) and d(GH)(S-2, S-3). We also formulate a number of open questions.