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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
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Category: math
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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
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Date available: 2009-01-08T18:04:50Z
Last updated: 2012-04-30
Stable URL: http://hdl.handle.net/10338.dmlcz/118598
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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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