  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.
