  Deng-Ning-Wang-Zhou showed that a Hermitian holomorphic vector bundle is
Griffiths semi-positive if it satisfies the optimal $L^2$-extension condition.
As a generalization, we present a quantitative characterization of Griffiths
positivity in terms of certain $L^2$-extension conditions. We also show that a
$\mathbb{R}$-valued measurable function is pluriharmonic if and only if it
satisfies the equality part of the optimal $L^p$-extension condition. This
answers a conjecture of Inayama affirmatively. Moreover, the flatness of a
possibly singular Hermitian metric is also equivalent to the equality part of
the optimal $L^p$-extension condition.
