= Zero Bregman distance and supporting faces
For $p\in\partial J(v)$, zero <Bregman distance> means that $J(u)$ equals the supporting affine function $J(v)+\langle p,u-v\rangle$. If $J$ is <strictly convex>, two distinct contact points are impossible: convexity along their segment would be strict, while the <subgradient inequality> supplies the opposite bound. Thus zero distance then implies $u=v$. Without strict convexity, $J(s)=|s|$, $v=1$, $p=1$ gives zero distance for every $u\geq0$. These are all on the same supporting face.
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