  We employ a basic formalism from convex analysis to show a simple relation
between the entanglement of formation $E_F$ and the conjugate function $E^*$ of
the entanglement function $E(\rho)=S(\trace_A\rho)$. We then consider the
conjectured strong superadditivity of the entanglement of formation $E_F(\rho)
\ge E_F(\rho_I)+E_F(\rho_{II})$, where $\rho_I$ and $\rho_{II}$ are the
reductions of $\rho$ to the different Hilbert space copies, and prove that it
is equivalent with subadditivity of $E^*$. As an application, we show that
strong superadditivity would follow from multiplicativity of the maximal
channel output purity for all non-trace-preserving quantum channels, when
purity is measured by Schatten $p$-norms for $p$ tending to 1.
