ABSTRACT
The idea of convexity finds its roots in the ideas of the ancient Greeks. What is it about convexity that is compelling, that makes us want to study the concept?
First of all, it is an elementary geometric idea that has immediate appeal. Second, it is based on positivity (at least as we present it here), and positivity is one of the most powerful ideas in mathematical analysis. Third, it is a concept that is stable under linear transformations and other elementary operations. Fourth, and perhaps most important, it is a notion of nonlinearity that is manageable and understandable.
