  We introduce the space of mixed-volume forms endowed with an $L^2$ metric on
a balanced manifold. A geodesic equation can be derived in this space that has
an interesting structure and extends the equation of Donaldson
\cite{Donaldson10} and Chen-He \cite{CH11} in the space of volume forms on a
Riemannian manifold. This nonlinear PDE is studied in detail and the existence
of weak solution is shown for the Dirichlet problem, under a positivity
assumption. Later we study the Calabi-Yau equation for balanced metrics and
introduce a geometric criteria for prescribing volume forms that is closely
related to the positivity assumption above. We call this assumption the
sub-Astheno-K\"ahler condition. By deriving $C^0$ a priori estimates, we show
that the existence of solutions can be established on all sub-Astheno-K\"ahler
manifolds.
