Reduced conditions for being Alexandrov spaces and applications

In this paper, we show that, in the definition of Alexandrov spaces with lower or upper curvature bound, the original conditions can be reduced to weaker ones. For the purpose, we introduce “imaginary” comparison angles (and “imaginary” 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.