Self-scaled barriers for semidefinite programming

I am going to show that all self-scaled barriers for the

cone of symmetric positive semidefinite matrices are of the form

$X\mapsto -c_1\ln\det X +c_0$ for some constants $c_1$ > $0,c_0 \in$ \RN.

Equivalently one could state say that all such functions may be

obtained via a homothetic transformation of the universal barrier

functional for this cone. The result shows that there is a certain

degree of redundancy in the axiomatic theory of self-scaled barriers,

and hence that certain aspects of this theory can be simplified. All

relevant concepts will be defined. In particular I am going to give

a short introduction to the notion of self-concordance and the

intuitive ideas that motivate its definition.