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A geometric nonconvex conic optimization problem (COP) was recently proposed by Kim, Kojima and Toh asa unified framework for convex conic reformulation of a class of quadratic optimization problems and polynomial optimization problems. The nonconvex COP minimizes a linear function over the intersection of a nonconvex cone K, a convex subcone J of the convex hull coK of K, and an affine hyperplane with a normal vector H. Under the assumption co(K \cap J) = J, the original nonconvex COP in their paper was shown to be equivalently formulated as a convex conic program by replacing the constraint set with the intersection of J and the affine hyperplane. This paper further studies some remaining issues, not fully investigated there, such as the key assumption co(K \cap J) = J in the framework. More specifically, we provide three sets of necessary-sufficient conditions for the assumption. As an application, we propose a new wide class of quadratically constrained quadratic programs with multiple nonconvex equality and inequality constraints that can be solved exactly by their semidefinite relaxation.