The Gromov-Hausdorff distance between S^n and S^{n+1} with geodesic metrics is exactly one half of arccos(-1/(n+1)).
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The Gromov-Hausdorff Distance Between Consecutive Spheres
The Gromov-Hausdorff distance between S^n and S^{n+1} with geodesic metrics is exactly one half of arccos(-1/(n+1)).