  Quadratically constrained quadratic programs (QCQPs) are a highly expressive
class of nonconvex optimization problems. While QCQPs are NP-hard in general,
they admit a natural convex relaxation via the standard semidefinite program
(SDP) relaxation. In this paper we study when the convex hull of the epigraph
of a QCQP coincides with the projected epigraph of the SDP relaxation. We
present a sufficient condition for convex hull exactness and show that this
condition is further necessary under an additional geometric assumption. The
sufficient condition is based on geometric properties of $\Gamma$, the cone of
convex Lagrange multipliers, and its relatives $\Gamma_1$ and $\Gamma^\circ$.
