The Gromov-Hausdorff distances between Alexandrov spaces of curvature bounded below by 1 and the standard spheres

Abstract

Main result in the present paper is the following: If an n-dimensional Alexandrov spaces X ⁿ of curvature ≥ 1 has radius greater than Π - ε, then the Gromov-Hausdor. distance between X ⁿ and the standard sphere S ⁿ is less than τ(ε). Here, τ(ε) is an explicit positive function depending only on ε such that lim_ε→0 τ(ε) = 0. We prove this by using quasigeodesics on Alexandrov spaces.