Geodesic
On differentiability properties of Lipschitz functions on a Banach space with a Lipschitz uniformly Gâteaux differentiable bump function
Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997) no. 2, pp. 329-336.

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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.
Classification : 41A65, 46B20, 46G05
Keywords: Lipschitz function; G\^ateaux differentiability; uniformly G\^ateaux differentiable; bump function; Banach-Mazur game; $\sigma$-porous set 
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     author = {Zaj{\'\i}\v{c}ek, Lud\v{e}k},
     title = {On differentiability properties of {Lipschitz} functions on a {Banach} space with a {Lipschitz} uniformly {G\^ateaux} differentiable bump function},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {329--336},
     publisher = {mathdoc},
     volume = {38},
     number = {2},
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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, Tome 38 (1997) no. 2, pp. 329-336. http://geodesic.mathdoc.fr/item/CMUC_1997__38_2_a12/