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 limε→0 τ(ε) = 0. We prove this by using quasigeodesics on Alexandrov spaces.
"The Gromov-Hausdorff distances between Alexandrov spaces of curvature bounded below by 1 and the standard spheres." Tsukuba J. Math. 35 (1) 1 - 12, June 2011. https://doi.org/10.21099/tkbjm/1311081446