Vol. 44, No. 2, 1973

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Subgradients of a convex function obtained from a directional derivative

Peter Drummond Taylor

Vol. 44 (1973), No. 2, 739–747

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.

Mathematical Subject Classification 2000
Primary: 46G05
Received: 8 October 1971
Published: 1 February 1973
Peter Drummond Taylor