It is known that a cone K in \(\mathbb {R}^n\) is sub-dual if and only if the maximal angle of K is less than or equal to \(\frac{\pi }{2}\) . Further, it is easy to verify that for a self-dual cone K, the maximal angle of K is \(\frac{\pi }{2}\) . However, this condition is not sufficient. In this paper using the concepts of antipodal and critical pairs, we give a necessary and sufficient condition for a cone K in \(\mathbb {R}^n\) to be self-dual. As a consequence, we observe that for a self-dual cone K, the set of all antipodal pairs, Nash pairs, critical pairs, and the complementarity set are the same and form an n-dimensional manifold.
