New definitions of Alexandrov space and applications

Abstract: In this paper we show that, in the definition of Alexandrov spaces with lower or upper curvature bound, the original conditions can be replaced with much weaker ones. For the purpose, we introduce imaginary' comparison angles (andimaginary' angles), and the right or left bounded second derivative in the support sense. As applications, we provide new proofs for the Doubling Theorem, and the Globalization Theorem for complete or geodesic Alexandrov spaces with lower curvature bound.