  This is the second part of our work on Zariski decomposition structures,
where we compare two different volume type functions for curve classes. The
first function is the polar transform of the volume for ample divisor classes.
The second function captures the asymptotic geometry of curves analogously to
the volume function for divisors. We prove that the two functions coincide,
generalizing Zariski's classical result for surfaces to all varieties. Our
result confirms the log concavity conjecture of the first named author for
weighted mobility of curve classes in an unexpected way, via Legendre-Fenchel
type transforms. We also give a number of applications to birational geometry,
including a refined structure theorem for the movable cone of curves.
