The Gromov-Hausdorff distance between two metric spaces measures how far the
spaces are from being isometric. It has played an important and longstanding
role in geometry and shape comparison. More recently, it has been discovered
that the Gromov-Hausdorff distance between unit spheres equipped with the
geodesic metric has important connections to Borsuk-Ulam theorems and
Vietoris-Rips complexes.
We develop a discrete framework for obtaining upper bounds on the
Gromov-Hausdorff distance between spheres, and provide the first quantitative
bounds that apply to spheres of all possible pairs of dimensions. As a special
case, we determine the exact Gromov-Hausdorff distance between the circle and
any even-dimensional sphere, and determine the asymptotic behavior of the
distance from the 2-sphere to the k-sphere up to constants.Comment: 13 pages, 3 figure