On inverses of $\delta$-convex mappings
Duda, Jakub
Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001), p. 281-297 / Harvested from Czech Digital Mathematics Library
Access to full text
Full (PDF)

In the first part of this paper, we prove that in a sense the class of bi-Lipschitz $\delta$-convex mappings, whose inverses are locally $\delta$-convex, is stable under finite-dimensional $\delta$-convex perturbations. In the second part, we construct two $\delta$-convex mappings from $\ell_1$ onto $\ell_1$, which are both bi-Lipschitz and their inverses are nowhere locally $\delta$-convex. The second mapping, whose construction is more complicated, has an invertible strict derivative at $0$. These mappings show that for (locally) $\delta$-convex mappings an infinite-dimensional analogue of the finite-dimensional theorem about $\delta$-convexity of inverse mappings (proved in [7]) cannot hold in general (the case of $\ell_2$ is still open) and answer three questions posed in [7].

Publié le : 2001-01-01
Classification:  46G99,  47H99,  58C20,  90C48 
@article{119243,
     author = {Jakub Duda},
     title = {On inverses of $\delta$-convex mappings},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {42},
     year = {2001},
     pages = {281-297},
     zbl = {1053.47522},
     mrnumber = {1832147},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119243}
}

Duda, Jakub. On inverses of $\delta$-convex mappings. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) pp. 281-297. http://gdmltest.u-ga.fr/item/119243/

Alexandrov A.D. On surfaces represented as the difference of convex functions, Izvest. Akad. Nauk. Kaz. SSR 60, Ser. Math. Mekh. 3 (1949), 3-20 (in Russian). (1949) | MR 0048059

Alexandrov A.D. Surfaces represented by the differences of convex functions, Doklady Akad. Nauk SSSR (N.S.) 72 (1950), 613-616 (in Russian). (1950) | MR 0038092

Cepedello Boiso M. Approximation of Lipschitz functions by $\Delta$-convex functions in Banach spaces, Israel J. Math. 106 (1998), 269-284. (1998) | MR 1656905 | Zbl 0920.46010

Cepedello Boiso M. On regularization in superreflexive Banach spaces by infimal convolution formulas, Studia Math. 129 (1998), 3 265-284. (1998) | MR 1609659 | Zbl 0918.46014

Hartman P. On functions representable as a difference of convex functions, Pacific J. Math. 9 (1959), 707-713. (1959) | MR 0110773 | Zbl 0093.06401

Kopecká E.; Malý J. Remarks on delta-convex functions, Comment. Math. Univ. Carolinae 31.3 (1990), 501-510. (1990) | MR 1078484

Veselý L.; Zajíček L. Delta-convex mappings between Banach spaces and applications, Dissertationes Math. (Rozprawy Mat.) 289 (1989), 52 pp. (1989) | MR 1016045