Contingent on the existence of certain affine maps, necessary and sufficient conditions for the eigenvalues of all matrices belonging to the convex hull of two given matrices to lie in a subset of the complex plane are obtained. Such maps are identified for every balanced convex domain D with polygonal boundary and conclusive D-stability criteria are obtained. In the case when D is the open left half plane, the computational complexity of the new test is somewhat lower than that of a previously proposed criterion. For nonpolygonal balanced convex domains, conditions that are as close to being necessary and sufficient as desired may be obtained via a suitable approximation of these domains by polygonal ones.