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This paper is the second in a two-part series on the concavity-convexity of the distribution system security region (DSSR). In Part I, the concave-convex characteristics and mechanism of DSSR are revealed by observing many case grids. In Part II (this paper), the concave-convex principle of DSSR is revealed through mathematical deduction. Firstly, the relevant concepts of sub-regions are proposed, and a DSSR model based on the union of sub-regions is developed. Secondly, the concave-convex principle of DSSR is proposed. The proposed criteria show that the security region of a distribution system with a single load transfer path for each feeder is convex, while that with multiple load transfer paths for some feeders can be concave or convex. Thirdly, the proposed criteria are deduced mathematically. Fourthly, based on the criteria, a method is proposed to judge whether a DSSR is concave or convex. Fifthly, the proposed criteria and method are verified through several cases, and the DSSR concavity-convexity of real distribution systems in China is also analyzed. Finally, the applications of DSSR concavity-convexity are illustrated. This work lays a mathematical foundation for the research of the topological properties of DSSR and is also useful for the application of DSSR. • The DSSR mathematical model based on the union of sub-regions is developed. • Three criteria are proposed and proved, revealing the concave-convex principle of DSSR. • A judgment method for DSSR concavity-convexity is proposed based on these criteria. • The DSSR concavity-convexity of real distribution systems in China is analyzed. • The application values of DSSR concavity-convexity are demonstrated. [ABSTRACT FROM AUTHOR]