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 [4], implies a sufficient condition for a family of continuous convex functions on a barrelled normed linear space to be locally uniformly Lipschitz.
@article{118598,
author = {Libor Vesel\'y},
title = {The distance between subdifferentials in the terms of functions},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {34},
year = {1993},
pages = {419-424},
zbl = {0809.49016},
mrnumber = {1243073},
language = {en},
url = {http://dml.mathdoc.fr/item/118598}
}
Veselý, Libor. The distance between subdifferentials in the terms of functions. Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) pp. 419-424. http://gdmltest.u-ga.fr/item/118598/
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