Suppose R is a lower
semicontinuous convex function on a Banach space E. A new result is obtained
relating the directional derivatives of h and its subgradients: if l is a tangent line at
some point z in graph h then a hyperplane can be found in E × R which supports
epigraph h at a point close to z and almost contains l. This theorem is applied to get
a formula for the directional derivative of h at a point in terms of the derivatives in
the same direction of subgradients at nearby points. This formula is used to
obtain several known results including the maximal monotonicity of the
subdifferential of h and the uniqueness of h with a given sub-differential. The
main lemma takes a point z in a closed convex set C, and a bounded set X,
all in a Banach space E, and gives conditions under which there exists a
hyperplane which supports C at a point close to z and separates C and
X.
