 Symmetric Bregman distance

For $p_u\in \partial J(u)$ and $p_v\in \partial J(v)$, add the two Bregman distances to get $D_J^{p_v}(u,v)+D_J^{p_u}(v,u)=⟨p_u-p_v,u-v⟩\ge 0$. The choice of subgradients matters. This is generally not a metric: it need not separate points or satisfy a triangle inequality.