L. Zaj´ıˇ cek
On dierentiability properties of Lipschitz functions on a Ba- nach space with a Lipschitz uniformly G^ateaux dierentiable bump function
Comment.Math.Univ.Carolinae 38,2 (1997) 329-336.
Abstract: We improve a theorem of P.G. Georgiev and N.P. Zlateva on Gˆateaux differentiability of Lipschitz functions in a Banach space which admits a Lipschitz uniformly Gˆateaux differentiable bump function. In particular, our result implies the following theorem: Ifdis a distance function determined by a closed subsetA of a Banach spaceX with a uniformly Gˆateaux differentiable norm, then the set of points ofX\Aat whichdis not Gˆateaux differentiable is not only a first category set, but it is evenσ-porous in a rather strong sense.
Keywords: Lipschitz function, Gˆateaux differentiability, uniformly Gˆateaux dif- ferentiable, bump function, Banach-Mazur game,σ-porous set
AMS Subject Classification: Primary 46G05; Secondary 41A65
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