Abstract: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.
Keywords: Lipschitz function, G\^ateaux differentiability, uniformly G\^ateaux differentiable, bump function, Banach-Mazur game, $\sigma $-porous set
AMS Subject Classification: Primary 46G05; Secondary 41A65