On the self-duality and homogeneity of ellipsoidal cones

The class of ellipsoidal cones, as an important prototype in closed convex cones, covers several practical instances such as second-order cone, circular cone and elliptic cone. In natural feature, it belongs to the category of nonsymmetric cones because it is non-self-dual under standard inner product. Nonetheless, it can be converted to a second-order cone, which is symmetric, by a transformation and vice versa. Is it possible to make an ellipsoidal cone to become self-dual by defining new setting of inner product? Is the class of ellipsoidal cones homogeneous? We provide affirmative answers for these two questions in this paper. As byproducts, its special cases such as circular cone and elliptic cone can be tackled likewise.