Differentiability of cone-monotone functions on separable Banach space
HTML articles powered by AMS MathViewer
- by Jonathan M. Borwein, James V. Burke and Adrian S. Lewis PDF
- Proc. Amer. Math. Soc. 132 (2004), 1067-1076 Request permission
Abstract:
Motivated by applications to (directionally) Lipschitz functions, we provide a general result on the almost everywhere Gâteaux differentiability of real-valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by a convex cone with non-empty interior. This seemingly arduous restriction is useful, since it covers the case of directionally Lipschitz functions, and necessary. We show by way of example that most results fail more generally.References
- Yoav Benyamini and Joram Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. MR 1727673, DOI 10.1090/coll/048
- J. M. Borwein and R. Goebel, “On the nondifferentiability of cone-monotone functions in Banach spaces," CECM preprint 02:179, March 2002.
- Jonathan M. Borwein and Warren B. Moors, Null sets and essentially smooth Lipschitz functions, SIAM J. Optim. 8 (1998), no. 2, 309–323. MR 1618798, DOI 10.1137/S1052623496305213
- J. M. Borwein and X. Wang, “Cone-monotone functions, differentiability and continuity,” preprint, Simon Fraser University.
- J. V. Burke, A. S. Lewis, and M. L. Overton, Approximating subdifferentials by random sampling of gradients, Math. Oper. Res. 27 (2002), no. 3, 567–584. MR 1926659, DOI 10.1287/moor.27.3.567.317
- Yves Chabrillac and J.-P. Crouzeix, Continuity and differentiability properties of monotone real functions of several real variables, Math. Programming Stud. 30 (1987), 1–16. Nonlinear analysis and optimization (Louvain-la-Neuve, 1983). MR 874128, DOI 10.1007/bfb0121151
- R. Tyrrell Rockafellar and Roger J.-B. Wets, Variational analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317, Springer-Verlag, Berlin, 1998. MR 1491362, DOI 10.1007/978-3-642-02431-3
- Stanisław Saks, Theory of the integral, Second revised edition, Dover Publications, Inc., New York, 1964. English translation by L. C. Young; With two additional notes by Stefan Banach. MR 0167578
- Karl R. Stromberg, Introduction to classical real analysis, Wadsworth International Mathematics Series, Wadsworth International, Belmont, Calif., 1981. MR 604364
Additional Information
- Jonathan M. Borwein
- Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
- Email: jborwein@cecm.sfu.ca
- James V. Burke
- Affiliation: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195-4350
- Email: burke@math.washington.edu
- Adrian S. Lewis
- Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
- Email: aslewis@cecm.sfu.ca
- Received by editor(s): April 11, 2002
- Received by editor(s) in revised form: November 19, 2002
- Published electronically: July 14, 2003
- Additional Notes: The first author’s research was supported by NSERC and by the Canada Research Chair Programme. The second author’s research was supported by NSF DMS-9971852 & NIH P41-RR-12609. The third author’s research was supported by NSERC
- Communicated by: N. Tomczak-Jaegermann
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 1067-1076
- MSC (2000): Primary 26B25; Secondary 90C29
- DOI: https://doi.org/10.1090/S0002-9939-03-07149-1
- MathSciNet review: 2045422