The properties of the barrier F(x)=-log(det(x)), defined over the cone of squares of an Euclidean Jordan algebra, are analyzed using pure algebraic techniques. Furthermore, relating the Carathéodory number of a symmetric cone with the rank of an underlying Euclidean Jordan algebra, conclusions about the optimal parameter of F are suitably obtained. Namely, it is proved that the Carathéodory number of the cone of squares of an Euclidean Jordan algebra is equal to the rank of the algebra. Then, taking into account the result obtained by Osman Guler and Levent Tunçel (Characterization of the barrier parameter of homogeneous convex cones, Mathematical Programming, 81 (1998): 55-76), where it is stated that the Carathéodory number of a symmetric cone Q is the optimal parameter of the self-concordant barrier defined over Q, allows the conclusion that the rank of an underlying Jordan algebra is also the self-concordant barrier optimal parameter.
Cadernos de Matemática, Universidade de Aveiro, November - 2003, CM03/I-32