This note was motivated by a
paper of P. D. Taylor, which contains a simple proof of Rockafellar’s basic theorem
that the subdifferential map ∂f of a lower semicontinuous proper convex
function f on a Banach space is maximal monotone. Taylor based his proof
on a theorem which can be considered as a sharpening (for the epigraph
of a convex function) of a result (Lemma 1.1) concerning support points
and functions of convex sets due to Brondsted-Rockafellar and Phelps. It is
shown that Taylor’s theorem can be generalized somewhat, using related
methods. (It is shown, by an example, that there is a limitation on the extent
of generalization possible.) The theorem follows from a slightly technical
result (Proposition 1.3) which admits a dual version (Proposition 2.2). As
an application of Proposition 2.2, a short proof of Rockafellar’s theorem
relating the graph of (∂f∗)−1 to that of ∂f is given. The methods of this
paper yield a generalization (Corollary 1.9) of one of the density results of
Bishop-Phelps.
