These references may help? Or at least lead you to related literature.
João Gouveia, Joao, and Rekha Thomas. "Convex hulls of algebraic sets." In Handbook on Semidefinite, Conic and Polynomial Optimization, pp. 113-138. Springer, Boston, MA, 2012. Pre-pub arXiv version.
"The main feature of the technique is that all computations are done modulo the ideal generated by the polynomials defining the set to the convexified."
Ranestad, Kristian, and Bernd Sturmfels. "The convex hull of a variety." In Notions of Positivity and the Geometry of Polynomials, pp. 331-344. Springer, Basel, 2011. Pre-pub arXiv version.