Each lower semi-continuous
proper convex function f on a Banach space E defines a certain multivalued mapping
∂f from E to E∗ called the subdifferential of f. It is shown here that the
mappings arising this way are precisely the ones whose graphs are maximal
“cyclically monotone” relations on E × E∗, and that each of these is also a
maximal monotone relation. Furthermore, it is proved that ∂f determines f
uniquely up to an additive constant. These facts generally fail to hold when
E is not a Banach space. The proofs depend on establishing a new result
which relates the directional derivatives of f to the existence of approximate
subgradients.
