Convexity and Cone-Vexing

The idea of convexity feeds generation, separation, calculus, and approximation. Generation appears as duality; separation, as optimality; calculus, as representation; and approximation, as stability. This is an overview of the origin, evolution, and trends of convexity. Study of convexity in the Sobolev Institute was initiated by Leonid Kantorovich (1912–1986) and Alexandr Alexandrov (1912–1999). This talk is a part of their memory.