On a problem of Bauschke and Borwein

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Consider a differentiable convex function f: Rⁿ ⊃ dom f → R. The induced spectral function F is given by F=f ∘ λ, where λ: Mₙsa → Rⁿ is the eigenvalue map. Let us denote by Df and DF 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 Df jointly convex if and only if DF is?" In this short note we provide a negative answer to this question.