Fundamental regions in convex programming

Question

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?

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