  We introduce a class of functional analogs of the symmetric difference metric
on the space of coercive convex functions on $\mathbb{R}^n$ with
full-dimensional domain. We show that convergence with respect to these metrics
is equivalent to epi-convergence. Furthermore, we give a full classification of
all isometries with respect to some of the new metrics. Moreover, we introduce
two new functional analogs of the Hausdorff metric on the spaces of coercive
convex functions and super-coercive convex functions, respectively, and prove
equivalence to epi-convergence.
