Where should I begin the study of convexity of (semi-)algebraic sets? In other words, projective varieties defined by polynomials of complex variables.
The long-term goal is to study optimization in such algebraic sets.
I have heard of "convex algebraic geometry" but they focus on real algebraic sets. I want to look in the complex case.
The first questions I want to know is how to define convexity in this space, and then what are the conditions for an algebraic set in this space to be convex?
Are there some textbooks or book chapters relevant to this area?
Thanks in advance.