# Article

 Title: The distance between subdifferentials in the terms of functions  (English) Author: Veselý, Libor Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 34 Issue: 3 Year: 1993 Pages: 419-424 . Category: math . Summary: 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. Keyword: convex analysis Keyword: subdifferentials of convex functions Keyword: barrelled normed linear spaces MSC: 26B25 MSC: 46A08 MSC: 46N10 MSC: 49J52 MSC: 52A41 idZBL: Zbl 0809.49016 idMR: MR1243073 . Date available: 2009-01-08T18:04:50Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/118598 . Reference: [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 Reference: [2] Phelps R.R.: Convex Functions, Monotone Operators and Differentiability.Lecture Notes in Mathematics, Vol. 1364, Springer-Verlag, Berlin-New York-Heidelberg, 1989. Zbl 0921.46039, MR 0984602 Reference: [3] Roberts A.W., Varberg D.E.: Convex Functions.Academic Press, New York-San Francisco- London, 1973. Zbl 0289.26012, MR 0442824 Reference: [4] Veselý L.: Local uniform boundedness principle for families of $\varepsilon$-monotone operators.to appear. MR 1326107 .

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