On a problem of Bauschke and Borwein

Abstract

Consider a differentiable convex function $f:{\mathbb{R}}^n\supset \mathrm{dom}f\to \mathbb{R}.$ The induced spectral function $F$ is given by $F=f\circ \lambda ,$ where $\lambda :{\mathbf{M}}_n^{sa}\to {\mathbb{R}}^n$ 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.

arXiv commentThis paper has been withdrawn by the author because a gap has been observed in the proof of a cited theorem which was an essential ingredient of our argument