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 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 d GH .S 0 ; S n /, d GH .S m ; S 1 /, 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. 53C23 1. Introduction 3734 2. Preliminaries 3746 3. Some general lower bounds 3748 4. The proof of Theorem A 3751 5. A Borsuk-Ulam theorem and the proof of Theorem B 3756 6. The proofs of Propositions 1.16 and 1.20 3760 7. The proof of Proposition 1.18 3767 8. The proof of Proposition 1.19 3775 9. The Gromov-Hausdorff distance between spheres 3784 Appendix A. A succinct proof of Theorem G 3793 Appendix B. The distance between a sphere and an interval 3795