Let E be a holomorphic vector bundle on a projective manifold X such that det E is ample. We introduce three functionals Φ_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-Ampère integrals of Φ_P (Θ_{E,h}), where Θ_{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 ΦP 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 ⊗ (det E)^t , where t > −1/ rank E, can possibly be investigated by studying the infimum of exponents t for which the Yang-Mills differential system has a solution.