A REMARK ON MULTI-DIMENSIONAL ANDO-HIAI LOG MAJORIZATION ON SYMMETRIC CONES

Recently, Bhatia et al. [3] established a multivariable extension of the Ando-Hiai log majorization theorem in terms of the Karcher mean on the cone P-n of positive definite matrices. As is well-known, P-n is one of the two representative examples of a symmetric cone in optimization fields. The other counterpart of P-n is the second-order cone (or Lorentz cone). It is of rank 2. In this work, we present the corresponding multivariable Ando-Hiai log majorization theorem with respect to the Karcher mean on the symmetric cone of rank 2 including the second-order cone. In addition, the case of rank 3 is treated.