On differentiability properties of Lipschitz functions on a Banach space with a Lipschitz uniformly Gâteaux differentiable bump function

Luděk Zajíček

Commentationes Mathematicae Universitatis Carolinae (1997)

  • Volume: 38, Issue: 2, page 329-336
  • ISSN: 0010-2628

Abstract

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We improve a theorem of P.G. Georgiev and N.P. Zlateva on Gâteaux differentiability of Lipschitz functions in a Banach space which admits a Lipschitz uniformly Gâteaux 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 Gâteaux differentiable norm, then the set of points of X ∖ A at which d is not Gâteaux differentiable is not only a first category set, but it is even σ -porous in a rather strong sense.

How to cite

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Zajíček, Luděk. "On differentiability properties of Lipschitz functions on a Banach space with a Lipschitz uniformly Gâteaux differentiable bump function." Commentationes Mathematicae Universitatis Carolinae 38.2 (1997): 329-336. <http://eudml.org/doc/248077>.

@article{Zajíček1997,
abstract = {We improve a theorem of P.G. Georgiev and N.P. Zlateva on Gâteaux differentiability of Lipschitz functions in a Banach space which admits a Lipschitz uniformly Gâteaux 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 Gâteaux differentiable norm, then the set of points of $X\setminus A$ at which $d$ is not Gâteaux differentiable is not only a first category set, but it is even $\sigma $-porous in a rather strong sense.},
author = {Zajíček, Luděk},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Lipschitz function; Gâteaux differentiability; uniformly Gâteaux differentiable; bump function; Banach-Mazur game; $\sigma $-porous set; Lipschitz function; Gâteaux differentiability; uniformly Gâteaux differentiable; bump function; Banach-Mazur game; -porous set},
language = {eng},
number = {2},
pages = {329-336},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On differentiability properties of Lipschitz functions on a Banach space with a Lipschitz uniformly Gâteaux differentiable bump function},
url = {http://eudml.org/doc/248077},
volume = {38},
year = {1997},
}

TY - JOUR
AU - Zajíček, Luděk
TI - On differentiability properties of Lipschitz functions on a Banach space with a Lipschitz uniformly Gâteaux differentiable bump function
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1997
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 38
IS - 2
SP - 329
EP - 336
AB - We improve a theorem of P.G. Georgiev and N.P. Zlateva on Gâteaux differentiability of Lipschitz functions in a Banach space which admits a Lipschitz uniformly Gâteaux 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 Gâteaux differentiable norm, then the set of points of $X\setminus A$ at which $d$ is not Gâteaux differentiable is not only a first category set, but it is even $\sigma $-porous in a rather strong sense.
LA - eng
KW - Lipschitz function; Gâteaux differentiability; uniformly Gâteaux differentiable; bump function; Banach-Mazur game; $\sigma $-porous set; Lipschitz function; Gâteaux differentiability; uniformly Gâteaux differentiable; bump function; Banach-Mazur game; -porous set
UR - http://eudml.org/doc/248077
ER -

References

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  3. Georgiev P.G., Submonotone mappings in Banach spaces and differentiability of non-convex functions, C.R. Acad. Sci. Bulg. 42 (1989), 13-16. (1989) Zbl0715.49016MR1020610
  4. Georgiev P.G., The smooth variational principle and generic differentiability, Bull. Austral. Math. Soc. 43 (1991), 169-175. (1991) Zbl0717.49014MR1086731
  5. Georgiev P.G., Submonotone mappings in Banach spaces and applications, preprint. Zbl0898.46015MR1451845
  6. Georgiev P.G., Zlateva N.P., An application of the smooth variational principle to generic Gâteaux differentiability, preprint. 
  7. Zajíček L., Differentiability of the distance function and points of multi-valuedness of the metric projection in Banach space, Czechoslovak Math. J. 33(108) (1983), 292-308. (1983) MR0699027
  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. (1984) MR0744405
  9. Zajíček L., A note on σ -porous sets, Real Analysis Exchange 17 (1991-92), p.18. (1991-92) 
  10. Zajíček L., Products of non- σ -porous sets and Foran systems, submitted to Atti Sem. Mat. Fis. Univ. Modena. MR1428780
  11. Zelený M., The Banach-Mazur game and σ -porosity, Fund. Math. 150 (1996), 197-210. (1996) MR1405042
  12. Zhivkov N.V., Generic Gâteaux differentiability of directionally differentiable mappings, Rev. Roumaine Math. Pures Appl. 32 (1987), 179-188. (1987) Zbl0628.46044MR0889011
  13. Wee-Kee Tang, Uniformly differentiable bump functions, preprint. MR1421846

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