Abstract:For convex continuous functions $f,g$ defined respectively in neighborhoods of points $x,y$ in a normed linear space, a formula for the distance between $\partial f(x)$ and $\partial g(y)$ in terms of $f,g$ (i.e. without using the dual) is proved. Some corollaries, like a new characterization of the subdifferential of a continuous convex function at a point, are given. This, together with a theorem from , implies a sufficient condition for a family of continuous convex functions on a barrelled normed linear space to be locally uniformly Lipschitz.
Keywords: convex analysis, subdifferentials of convex functions, barrelled normed linear spaces
AMS Subject Classification: Primary 26B25, 52A41; Secondary 46A08