  Utilizing dual descriptions of the normal cone of convex optimization
problems in conic form, we characterize the vertices of semidefinite
representations arising from Lov\'asz theta body, generalizations of the
elliptope, and related convex sets. Our results generalize vertex
characterizations due to Laurent and Poljak from the 1990's. Our approach also
leads us to nice characterizations of strict complementarity and to connections
with some of the related literature.
