In this paper we define a weak (n+1,ε)−strainer on an Alexandrov space with curvature ≥ 1, and prove an almost isometric sphere theorem in the setting of a weak strainer, making use of a rigidity theorem f or round spheres. To prove the rigidity theorem we investigate several proper ties of weak strainers, e.g. the maximality property, the covering property of the balls centered at strainer points, and an equilibrium property of a maximal ly separated weak strainer. At last we study several… Expand

This paper is mainly devoted to proving the four equivalent defining properties of a CAT(κ) space. The proof is based on an interesting tool we established which describes the cyclical five-step… Expand

We give a proof of the celebrated stability theorem of Perelman stating that for a noncollapsing sequence Xi of Alexandrov spaces with curv > k Gromov-Hausdorff converging to a compact Alexandrov… Expand

After the seminal work of Gromov (see [G1],[GLP]), questions of this type, with various assumptions on curvatures, and other geometric characteristics, have been receiving much attention. Cheeger,… Expand