On Marginal Subgradients of Convex Functions

For a lower semi continuous 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 $\partial f(x)$ with the smallest norm. We denote the norm of the marginal subgradient of $f$ at $x$ by $g(x)$. In this paper we study 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.