An almost isometric sphere theorem and weak strainers on Alexandrov spaces

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 for round spheres. To prove the rigidity theorem we investigate several properties of weak strainers, e.g. the maximality property, the covering property of the balls centered at strainer points, and an equilibrium property of a maximally separated weak strainer. At last we study several properties of regular points related to weak strainers, and prove the openness of a tight map. Mathematics Subject Classification (2010). 53C23;53C24;52C17.