Friday, October 12, 2007 



On the Tunçel Conjecture: A New Class of SelfConcordant Barriers on Sets of Symmetric Matrices 

Given a separable strongly selfconcordant function $f:\R^n
\rightarrow \R$, we show the associated spectral function $F(X)= (f \circ
\lambda)(X)$ is also strongly selfconcordant function. In addition, there
is a universal constant $\mathcal{O}$ such that, if $f(x)$ is separable
selfconcordant barrier then $\mathcal{O}^2F(X)$ is a selfconcordant
barrier. We estimate that for the universal constant we have $\mathcal{O}
\le 22$. This generalizes the relationship between the standard
logarithmic barriers $\sum_{i=1}^n\log x_i$ and $\log \det X$ and gives
a partial solution to a conjecture of L. Tunçel.
