Abstract of CORC technical report TR-2000-01:
Complete Classification of Self-scaled Barrier Functions
Stefan H. Schmieta, 2000
Self-scaled barrier functions are an important tool in the study
of conic linear programming problems over symmetric cones. They
are also closely related to Euclidean Jordan algebras. In the first
part of this article we show that every self-scaled barrier function defines
a Euclidean Jordan algebra. In the second part we use this algebra
to obtain a complete classification of self-scaled barrier functions.
In particular we show that the barrier function with the smallest possible
self-concordance parameter is unique up to an additive constant and
can be written in terms of the determinant of the associated Jordan
algebra.