Porosity and differentiability in smooth Banach spaces

Georgiev, Pando

doi:10.1090/S0002-9939-05-07736-1

Porosity and differentiability in smooth Banach spaces
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Proc. Amer. Math. Soc. 133 (2005), 1621-1628 Request permission

Abstract:

We improve a result of Preiss, Phelps and Namioka, showing that every submonotone mapping in a Gateaux smooth Banach space is single-valued on the complement of a $\sigma$-cone porous subset. If a Banach space $E$ has a uniformly $\beta$-differentiable Lipschitz bump function (with respect to some bornology $\beta$), then we show with a much simpler argument (localization of $\delta$-minimum of a perturbed function) that every continuous convex function on $E$ is $\beta$-differentiable on the complement of a $\sigma$-uniformly porous set.

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