On a problem of Bauschke and Borwein

Consider a differentiable convex function $f: $\mathbb{R}$^n \supset $\mathrm{d}\mathrm{o}\mathrm{m}$ f \rightarrow $\mathbb{R}$.$Theinducedspectralfunction$F$ is given by $F=f\circ \lambda ,$ where $\lambda: $\mathbf{M}$_n^{sa} \rightarrow $\mathbb{R}$^{n}$istheeigenvaluemap\mathrm{.}Letusdenoteby$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.