Title: Subgradients of a convex function obtained from a directional derivative
Abstract: 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 I is a tangent line at some point z in graph h then a hyperplane can be found in E X R which supports epigraph h at a point close to z and almost contains I.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 subdifferential.The main lemma takes a point nn 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 Cat a point close to z and separates C and X.A proper convex function on a real Banach space E is a function h on E with values in (-°°, +°°), not identically +°o, such thatfor x,y eE and 0 ^ λ <^ 1.A subgradient of h at x e E is an s e E* such that h(y) ^ h(x)s(yx) for all y eE .