  Having studied families of antiderivatives and their envelopes in the setting
of classical convex analysis, we now extend and apply these notions and results
in settings of abstract convex analysis. Given partial data regarding a
c-subdifferential, we consider the set of all c-convex c-antiderivatives that
comply with the given data. Under a certain assumption, this set is not empty
and contains both its lower and upper envelopes. We represent these optimal
antiderivatives by explicit formulae. Some well known functions are, in fact,
optimal c-convex c-antiderivatives. In one application, we point out a natural
minimality property of the Fitzpatrick function of a c-monotone mapping, namely
that it is a minimal antiderivative. In another application, in metric spaces,
a constrained Lipschitz extension problem fits naturally the convexity notions
we discuss here. It turns out that the optimal Lipschitz extensions are
precisely the optimal antiderivatives. This approach yields explicit formulae
for these extensions, the most particular case of which recovers the well known
extensions due to McShane and Whitney.
