Quantitative upper bounds on the Gromov-Hausdorff distance between spheres

TL;DR

This paper develops a discrete framework to establish quantitative upper bounds on the Gromov-Hausdorff distance between spheres of various dimensions, including exact distances for specific cases and asymptotic behavior.

Contribution

It introduces the first comprehensive quantitative bounds for the Gromov-Hausdorff distance between spheres of all dimensions, including exact and asymptotic results.

Findings

Exact Gromov-Hausdorff distance between circle and higher-dimensional spheres.

Asymptotic behavior of the distance from 2-sphere to k-sphere.

Development of a discrete framework for bounding distances.

Abstract

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 higher-dimensional sphere, and determine the precise asymptotic behavior of the distance from the 2-sphere to the $k$-sphere up to constants.