  Let $k \ge 1$ be an integer and $f$ a holomorphic endomorphism of $\mathbb
P^k (\mathbb C)$ of algebraic degree $d\geq 2$. We introduce a volume dimension
for ergodic $f$-invariant probability measures with strictly positive Lyapunov
exponents. In particular, this class of measures includes all ergodic measures
whose measure-theoretic entropy is strictly larger than $(k-1)\log d$, a
natural generalization of the class of measures of positive measure-theoretic
entropy in dimension 1. The volume dimension is equivalent to the Hausdorff
dimension when $k=1$, but depends on the dynamics of $f$ to incorporate the
possible failure of Koebe's theorem and the non-conformality of holomorphic
endomorphisms for $k\geq 2$.
  If $\nu$ is an ergodic $f$-invariant probability measure with strictly
positive Lyapunov exponents, we prove a generalization of the Ma\~n\'e-Manning
formula relating the volume dimension, the measure-theoretic entropy, and the
sum of the Lyapunov exponents of $\nu$. As a consequence, we give a
characterization of the first zero of a natural pressure function for such
expanding measures in terms of their volume dimensions. For hyperbolic maps,
such zero also coincides with the volume dimension of the Julia set, and with
the exponent of a natural (volume-)conformal measure. This generalizes results
by Denker-Urba\'nski and McMullen in dimension 1 to any dimension $k\geq 1$.
  Our methods mainly rely on a theorem by Berteloot-Dupont-Molino, which gives
a precise control on the distortion of inverse branches of endomorphisms along
generic inverse orbits with respect to measures with strictly positive Lyapunov
exponents.
