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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Generalized subdifferentials: a Baire categorical approach


Authors: Jonathan M. Borwein, Warren B. Moors and Xianfu Wang
Journal: Trans. Amer. Math. Soc. 353 (2001), 3875-3893
MSC (1991): Primary 49J52, 54E52
Published electronically: May 14, 2001
MathSciNet review: 1837212
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Abstract: We use Baire categorical arguments to construct pathological locally Lipschitz functions. The origins of this approach can be traced back to Banach and Mazurkiewicz (1931) who independently used similar categorical arguments to show that ``almost every continuous real-valued function defined on [0,1] is nowhere differentiable". As with the results of Banach and Mazurkiewicz, it appears that it is easier to show that almost every function possesses a certain property than to construct a single concrete example. Among the most striking results contained in this paper are: Almost every 1-Lipschitz function defined on a Banach space has a Clarke subdifferential mapping that is identically equal to the dual ball; if $\{T_{1}, T_{2},\ldots, T_{n}\}$ is a family of maximal cyclically monotone operators defined on a Banach space $X$ then there exists a real-valued locally Lipschitz function $g$such that $\partial_{0}g(x)=\mbox{co}\{T_{1}(x),T_{2}(x),\ldots, T_{n}(x)\}$for each $x\in X$; in a separable Banach space each non-empty weak$^{*}$compact convex subset in the dual space is identically equal to the approximate subdifferential mapping of some Lipschitz function and for locally Lipschitz functions defined on separable spaces the notions of strong and weak integrability coincide.


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Additional Information

Jonathan M. Borwein
Affiliation: Centre for Experimental and Constructive Mathematics, Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C. V5A1S6, Canada
Email: jborwein@cecm.sfu.ca

Warren B. Moors
Affiliation: Department of Mathematics, The University of Waikato, Private bag 3105 Hamilton, New Zealand
Email: moors@math.waikato.ac.nz

Xianfu Wang
Affiliation: Centre for Experimental and Constructive Mathematics, Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C. V5A1S6, Canada
Email: xwang@cecm.sfu.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02820-3
PII: S 0002-9947(01)02820-3
Keywords: Subdifferentials, differentiability, Baire category, upper semi--continuous set--valued map, $T$-Lipschitz function
Received by editor(s): March 24, 1999
Received by editor(s) in revised form: February 25, 2000
Published electronically: May 14, 2001
Additional Notes: Research of the first author was supported by NSERC and the Shrum endowment of Simon Fraser University
Research of the second author was supported by a Marsden fund grant, VUW 703, administered by the Royal Society of New Zealand
Article copyright: © Copyright 2001 American Mathematical Society