Fundamental regions of linear programming are polyhedra (since those are the objects of intersection of linear inequalities) and for semidefinite programming it is spectrahedra (https://math.berkeley.edu/~bernd/coimbra1.pdf says A spectrahedron is the intersection of the cone of positive semidefinite matrices with an affine-linear space).

What are the fundamental regions in convex programming in general?

In linear programming, the fundamental regions are polyhedra, since those are the intersection of half-spaces defined by linear inequalities. In semidefinite programming, the fundamental regions are spectrahedra$^\color{magenta}{\dagger}$:

A spectrahedron is the intersection of the cone of positive semidefinite matrices with an affine-linear space.

What are the fundamental regions in convex programming in general?

_{$\color{magenta}{\dagger}$ Bernd Sturmfels, GAeL Lecture I on Convex Algebraic Geometry — Spectrahedra, Coimbra, June 2010.}