Every compact convex subset of matrices is the Clarke Jacobian of some Lipschitzian mapping

Bartl, David; Fabian, Marián

doi:10.1090/proc/15571

Every compact convex subset of matrices is the Clarke Jacobian of some Lipschitzian mapping
HTML articles powered by AMS MathViewer

by David Bartl and Marián Fabian PDF

Proc. Amer. Math. Soc. 149 (2021), 4771-4779 Request permission

Abstract:

We prove that every non-empty compact convex subset of $m\times n$ matrices is the Clarke Jacobian of a Lipschitzian mapping from $\mathbb {R}^n$ to $\mathbb {R}^m$.

References

David Bartl and Marián Fabian, Can Pourciau’s open mapping theorem be derived from Clarke’s inverse mapping theorem easily?, J. Math. Anal. Appl. 497 (2021), no. 2, Paper No. 124858, 13. MR 4196569, DOI 10.1016/j.jmaa.2020.124858
Jonathan M. Borwein, Warren B. Moors, and Xianfu Wang, Lipschitz functions with prescribed derivatives and subderivatives, Nonlinear Anal. 29 (1997), no. 1, 53–63. MR 1447569, DOI 10.1016/S0362-546X(96)00050-8
Jonathan M. Borwein, Warren B. Moors, and Xianfu Wang, Generalized subdifferentials: a Baire categorical approach, Trans. Amer. Math. Soc. 353 (2001), no. 10, 3875–3893. MR 1837212, DOI 10.1090/S0002-9947-01-02820-3
M. Bivas, A. Daniilidis, and M. Quincampoix, Characterization of Filippov representable maps and Clarke subdifferentials, Math. Program, 2020. https://doi.org/10.1007/s10107-020-01540-y
Frank H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 709590
A. F. Izmailov, On a problem of existence of a nondegeneracy subspace for a convex compact family of epimorphisms. In Russian. In: Bereznev, V.A. (ed.) Theoretical and Applied Problems of Nonlinear Analysis, Computer Center RAS, Moscow (2010), 34–49.
Serge Lang, Linear algebra, 3rd ed., Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1987. MR 874113, DOI 10.1007/978-1-4757-1949-9
Zsolt Páles and Vera Zeidan, Infinite dimensional generalized Jacobian: properties and calculus rules, J. Math. Anal. Appl. 344 (2008), no. 1, 55–75. MR 2416293, DOI 10.1016/j.jmaa.2008.02.044
T. H. Sweetser III, A minimal set-valued strong derivative for vector-valued Lipschitz functions, J. Optim. Theory Appl. 23 (1977), no. 4, 549–562. MR 468174, DOI 10.1007/BF00933296