| Title:
|
On differentiability properties of Lipschitz functions on a Banach space with a Lipschitz uniformly Gâteaux differentiable bump function (English) |
| Author:
|
Zajíček, Luděk |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
38 |
| Issue:
|
2 |
| Year:
|
1997 |
| Pages:
|
329-336 |
| . |
| Category:
|
math |
| . |
| Summary:
|
We improve a theorem of P.G. Georgiev and N.P. Zlateva on G\^ateaux differentiability of Lipschitz functions in a Banach space which admits a Lipschitz uniformly G\^ateaux differentiable bump function. In particular, our result implies the following theorem: If $d$ is a distance function determined by a closed subset $A$ of a Banach space $X$ with a uniformly G\^ateaux differentiable norm, then the set of points of $X\setminus A$ at which $d$ is not G\^ateaux differentiable is not only a first category set, but it is even $\sigma$-porous in a rather strong sense. (English) |
| Keyword:
|
Lipschitz function |
| Keyword:
|
G\^ateaux differentiability |
| Keyword:
|
uniformly G\^ateaux differentiable |
| Keyword:
|
bump function |
| Keyword:
|
Banach-Mazur game |
| Keyword:
|
$\sigma$-porous set |
| MSC:
|
41A65 |
| MSC:
|
46B20 |
| MSC:
|
46G05 |
| idZBL:
|
Zbl 0886.46049 |
| idMR:
|
MR1455499 |
| . |
| Date available:
|
2009-01-08T18:31:11Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118930 |
| . |
| Reference:
|
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| Reference:
|
[2] Fabian M., Zhivkov N.V.: A characterization of Asplund spaces with the help of local $\epsilon$-supports of Ekeland and Lebourg.C.R. Acad. Sci. Bulg. 38 (1985), 671-674. Zbl 0577.46012, MR 0805439 |
| Reference:
|
[3] Georgiev P.G.: Submonotone mappings in Banach spaces and differentiability of non-convex functions.C.R. Acad. Sci. Bulg. 42 (1989), 13-16. Zbl 0715.49016, MR 1020610 |
| Reference:
|
[4] Georgiev P.G.: The smooth variational principle and generic differentiability.Bull. Austral. Math. Soc. 43 (1991), 169-175. Zbl 0717.49014, MR 1086731 |
| Reference:
|
[5] Georgiev P.G.: Submonotone mappings in Banach spaces and applications.preprint. Zbl 0898.46015, MR 1451845 |
| Reference:
|
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| Reference:
|
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| Reference:
|
[8] Zajíček L.: A generalization of an Ekeland-Lebourg theorem and the differentiability of distance functions.Suppl. Rend. Circ. Mat. di Palermo, Ser. II 3 (1984), 403-410. MR 0744405 |
| Reference:
|
[9] Zajíček L.: A note on $\sigma$-porous sets.Real Analysis Exchange 17 (1991-92), p.18. |
| Reference:
|
[10] Zajíček L.: Products of non-$\sigma$-porous sets and Foran systems.submitted to Atti Sem. Mat. Fis. Univ. Modena. MR 1428780 |
| Reference:
|
[11] Zelený M.: The Banach-Mazur game and $\sigma$-porosity.Fund. Math. 150 (1996), 197-210. MR 1405042 |
| Reference:
|
[12] Zhivkov N.V.: Generic Gâteaux differentiability of directionally differentiable mappings.Rev. Roumaine Math. Pures Appl. 32 (1987), 179-188. Zbl 0628.46044, MR 0889011 |
| Reference:
|
[13] Wee-Kee Tang: Uniformly differentiable bump functions.preprint. MR 1421846 |
| . |
