Convex Hulls of Quadratically Parameterized Sets With Quadratic Constraints

Let V be a semialgebraic set parameterized by quadratic polynomials over a quadratic set T. This paper studies semidefinite representation of its convex hull by projections of spectrahedra (defined by linear matrix inequalities). When T is defined by a single quadratic constraint, we prove that its convex hull is equal to the first order moment type semidefinite relaxation of $V$, up to taking closures. Similar results hold when every quadratic polynomial is homogeneous and T is defined by two homogeneous quadratic constraints,or V is defined by rational quadratic parameterizations.