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

Abstract

In this paper we give a new proof (along the line of the original proof) for the almost isometry theorem in Alexandrov geometry with curvature bounded below in Yu. Burago, M. Gromov, and G. Perel’man, A. D. Alexandrov spaces with curvature bounded below, Russian Math. Surveys, 47:2 (1992), pp. 1–58. The motivation of the new proof is that we find that Lemma 9.11 in A. D. Alexandrov spaces with curvature bounded below is incorrect (see Example 1.3 below), while this lemma is a crucial step in the original proof.