  Let $E$ be a holomorphic vector bundle on a projective manifold $X$ such that
$\det E$ is ample. We introduce three functionals $\Phi_P$ related to
Griffiths, Nakano and dual Nakano positivity respectively. They can be used to
define new concepts of volume for the vector bundle $E$, by means of
generalized Monge-Amp\`ere integrals of $\Phi_P(\Theta_{E,h})$, where
$\Theta_{E,h}$ is the Chern curvature tensor of $(E,h)$. These volumes are
shown to satisfy optimal Chern class inequalities. We also prove that the
functionals $\Phi_P$ give rise in a natural way to elliptic differential
systems of Hermitian-Yang-Mills type for the curvature, in such a way that the
related $P$-positivity threshold of $E\otimes(\det E)^t$, where $t>-1/{\rm
rank} E$, can possibly be investigated by studying the infimum of exponents $t$
for which the Yang-Mills differential system has a solution.
