On Marginal Subgradients of Convex Functions

For a lower semicontinuous and proper convex function f with nonempty minimizer set and a point x in its domain, a marginal subgradient of f at x is a vector in ∂ f(x) with the smallest norm. We denote the norm of the marginal subgradient of f at x by g(x), it is known that the infimum of g(x) over a level set of f is nondecreasing from a lower level set to a higher level set. In this paper we study another aspect of the marginal subgradients, namely the monotonicity of the infimum of g(x) over an equidistance contour from the minimizer set. The results are applied to the study of some growth properties of the marginal subgradients.