The Gromov-Hausdorff distances between Alexandrov spaces of curvature bounded below by 1 and the standard spheres (Q638742)

Let \(X^n\) be an \(n\)-dimensional Alexandrov space of curvature bounded from below by \(1\). The diameter of such a space cannot exceed \(\pi\) [\textit{Yu. Burago, M. Gromov} and \textit{G. Perel'man}, Russ. Math. Surv. 47, No. 2, 1--58; translation from Usp. Mat. Nauk 47, No. 2 (284), 3--51 (1992; Zbl 0802.53018)], and therefore the radius of \(X^n\) is also bounded by \(\pi\). The radius bound is attained only by the standard sphere \(S^n\). Moreover, if \(\mathrm{rad}\,X^n>\pi/2\), then \(X^n\) is homeomorphic to \(S^n\), see [\textit{K. Grove} and \textit{P. Petersen}, Invent. Math. 112, No. 3, 577--583 (1993; Zbl 0801.53029)]. Grove and Petersen also proved the following stability result: There exists \(\epsilon_n>0\) such that \(\mathrm{rad}\,X^n>\pi-\epsilon_n\) implies that \(X^n\) is bi-Lipschitz equivalent to \(S^n\). The main result of the present article is a dimension-independent stability theorem in terms of the Gromov-Hausdorff distance \(d_{GH}\). Namely, it states that for every \(\tau>0\) there exists \(\epsilon>0\), independent of \(n\), such that \(\mathrm{rad}\,X^n>\pi-\epsilon\) implies \(d_{GH}(X^n,S^n)<\tau\).