A new proof of almost isometry theorem in Alexandrov geometry with curvature bounded below (Q2446028)

This paper gives a new proof of the almost isometry theorem for Alexandrov spaces of \textit{Yu. Burago} et al. [Russ. Math. Surv. 47, No. 2, 1--58 (1992); translation from Usp. Mat. Nauk 47, No. 2(284), 3--51 (1992; Zbl 0802.53018)], following essentially the same lines and correcting a wrong lemma. The almost isometry theorem asserts that \(n\)-dimensional Alexandrov spaces with the same curvature bounds that are close for the Gromov-Hausdorff distance are quasi-isometric. The theorem must be stated with quantifiers. The wrong lemma in the original paper deals with the center of mass of finitely many points, which is the main tool used for constructing quasi-isometries from Gromov-Hausdorff approximations. The authors provide a counterexample to the original statement, give a new lemma and apply it to the proof of the theorem, with improved quantifiers.