  I calculate the 1-loop vacuum polarization $\Pi_{\mu\nu}(k,B,a)$ for a photon
of momentum $k=(\hat k,k_3)$ interacting with the electrons of a thin medium of
thickness $2a$ simulating graphene, in the presence of a constant and uniform
external magnetic field $B$ orthogonal to it (parallel to $k_3$). Calculations
are done with the techniques of Schwinger, adapted to the geometry and
Hamiltonian under scrutiny. The situation gets more involved than for the
electron self-energy because the photon is now allowed to also propagate
outside the medium. This makes $\Pi_{\mu\nu}$ factorize into a quantum,
"reduced" $T_{\mu\nu}(\hat k,B)$ and a transmittance function $V(k,a)$, in
which the geometry of the sample and the resulting confinement of the
$\gamma\,e^+\,e^-$ vertices play major roles. This drags the results away from
reduced QED$_{3+1}$ on a 2-brane. The finiteness of $V$ at $k^2=0$ is an
essential ingredient to fulfill suitable renormalization condition for
$\Pi_{\mu\nu}$ and to fix the corresponding counterterms. Their connection with
the transversality of $\Pi_{\mu\nu}$ is investigated. The corrections to the
Coulomb potential and their dependence on $B$ strongly differ from QED$_{3+1}$.
