On a problem of Bauschke and Borwein

Abstract

Consider a differentiable convex function $f: \mathbb{R}^n \supset \mathrm{dom} f \rightarrow \mathbb{R}.$ The induced spectral function $F$ is given by $F=f \circ \lambda,$ where $\lambda: \mathbf{M}_n^{sa} \rightarrow \mathbb{R}^{n}$ is the eigenvalue map. Let us denote by $D_f$ and $D_F$ the Bregman distances associated with $f$ and $F,$ respectively. In the paper "Joint and separate convexity of the Bregman distance" written by H. Bauschke and J. Borwein the following open problem has been suggested. "Is $D_f$ jointly convex if and only if $D_F$ is?" In this short note we provide a negative answer to this question.