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Porosity and differentiability in smooth Banach spaces


Author: Pando Gr. Georgiev
Journal: Proc. Amer. Math. Soc. 133 (2005), 1621-1628
MSC (2000): Primary 49J53; Secondary 49J50
DOI: https://doi.org/10.1090/S0002-9939-05-07736-1
Published electronically: January 14, 2005
MathSciNet review: 2120263
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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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Additional Information

Pando Gr. Georgiev
Affiliation: Department of Mathematics and Informatics, Sofia University “St. Kl. Ohridski", 5 James Bourchier Boulevard, 1126 Sofia, Bulgaria
Address at time of publication: Electrical & Computer Engineering and Computer Science Department, University of Cincinnati, ML 0030, Cincinnati, Ohio 45220
Email: pgeorgie@ececs.uc.edu

DOI: https://doi.org/10.1090/S0002-9939-05-07736-1
Keywords: Porous set, submonotone mappings, differentiability
Received by editor(s): July 31, 2002
Published electronically: January 14, 2005
Dedicated: Dedicated to Professor Petar Kenderov on the occasion of his 60th anniversary
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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