Title:

The distance between subdifferentials in the terms of functions
(English)

Author:

Veselý, Libor 
Language:

English 
Journal:

Commentationes Mathematicae Universitatis Carolinae 
ISSN:

00102628 (print) 
ISSN:

12137243 (online) 
Volume:

34 
Issue:

3 
Year:

1993 
Pages:

419424 
. 
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:

20090108T18:04:50Z 
Last updated:

20120430 
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, BostonLondonMelbourne, 1982. MR 0650456 
Reference:

[2] Phelps R.R.: Convex Functions, Monotone Operators and Differentiability.Lecture Notes in Mathematics, Vol. 1364, SpringerVerlag, BerlinNew YorkHeidelberg, 1989. Zbl 0921.46039, MR 0984602 
Reference:

[3] Roberts A.W., Varberg D.E.: Convex Functions.Academic Press, New YorkSan 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 
. 