 Zero Bregman distance and supporting faces

For $p\in \partial J(v)$, zero Bregman distance means that $J(u)$ equals the supporting affine function $J(v)+⟨p,u-v⟩$. 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 $u\ge 0$. These are all on the same supporting face.