Self-scaled Barriers for Semidefinite Programming

We show a result that can be expressed in any of the following three equivalent ways: 1. All self-scaled barrier functionals for the cone + of symmetric positive semideenite matrices are homothetic transformation of the universal barrier functional. 2. All self-scaled barrier functionals for + can be expressed in the form X 7 ! ?c 1 ln det X + c 0 for some constants c 1 > 0; c 0 2 R. 3. All self-scaled barrier functionals for + are isotropic. As a consequence we nd that a self-concordant barrier functional H for + is self-scaled if and only if Aut((+) acts as a group of translations on H , and that the closed subgroup of Aut((+) generated by the set of Hessians of a self-scaled barrier H coincides with the orientation preserving part of Aut((+).