| 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. (English) |
| 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 |
| . |
