Spectral Self-Concordant Barriers: A Proof of a Conjecture by Tunçel

Let $Q\subseteq {\mathbb{R}}^n$ be a open, convex, and symmetric set and let $f:Q\to \mathbb{R}$ be a $\vartheta$-self-concordant barrier function, that is also symmetric. We show the associated spectral function $F(X)={\mathcal{O}}^2(f\circ \lambda )(X)$ is a ${\mathcal{O}}^2\vartheta$-self-concordant function, where $\mathcal{O}=8\max \{1+3\vartheta ,12\}$, on the domain of all $n\times n$ symmetric matrices $X$, with eigenvalues $\lambda (X)\in Q$. This generalizes the relationship between the standard logarithmic barriers $-{\sum }_{i=1}^n\log x_i$ and $-\log \det X$ and gives a solution to a conjecture by L.~Tunçel, 1995.