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Keywords:
Lipschitz function; G\^ateaux differentiability; uniformly G\^ateaux differentiable; bump function; Banach-Mazur game; $\sigma$-porous set
Lipschitz function; G\^ateaux differentiability; uniformly G\^ateaux differentiable; bump function; Banach-Mazur game; $\sigma$-porous set
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.
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