  This is the first part of our work on Zariski decomposition structures, where
we study Zariski decompositions using Legendre-Fenchel type transforms. In this
way we define a Zariski decomposition for curve classes. This decomposition
enables us to develop the theory of the volume function for curves defined by
the second named author, yielding some fundamental positivity results for curve
classes. For varieties with special structures, the Zariski decomposition for
curve classes admits an interesting geometric interpretation.
