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convex analysis; subdifferentials of convex functions; barrelled normed linear spaces
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.
[1] Giles J.R.: Convex Analysis with Application in Differentiation of Convex Functions. Research Notes in Mathematics, Vol. 58, Pitman, Boston-London-Melbourne, 1982. MR 0650456
[2] Phelps R.R.: Convex Functions, Monotone Operators and Differentiability. Lecture Notes in Mathematics, Vol. 1364, Springer-Verlag, Berlin-New York-Heidelberg, 1989. MR 0984602 | Zbl 0921.46039
[3] Roberts A.W., Varberg D.E.: Convex Functions. Academic Press, New York-San Francisco- London, 1973. MR 0442824 | Zbl 0289.26012
[4] Veselý L.: Local uniform boundedness principle for families of $\varepsilon$-monotone operators. to appear. MR 1326107
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