| Title:
|
On inverses of $\delta$-convex mappings (English) |
| Author:
|
Duda, Jakub |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
42 |
| Issue:
|
2 |
| Year:
|
2001 |
| Pages:
|
281-297 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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]. (English) |
| Keyword:
|
delta-convex mappings |
| Keyword:
|
strict differentiability |
| Keyword:
|
normed linear spaces |
| MSC:
|
46G99 |
| MSC:
|
47H99 |
| MSC:
|
58C20 |
| MSC:
|
90C48 |
| idZBL:
|
Zbl 1053.47522 |
| idMR:
|
MR1832147 |
| . |
| Date available:
|
2009-01-08T19:09:54Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119243 |
| . |
| Reference:
|
[1] 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). MR 0048059 |
| Reference:
|
[2] Alexandrov A.D.: Surfaces represented by the differences of convex functions.Doklady Akad. Nauk SSSR (N.S.) 72 (1950), 613-616 (in Russian). MR 0038092 |
| Reference:
|
[3] Cepedello Boiso M.: Approximation of Lipschitz functions by $\Delta$-convex functions in Banach spaces.Israel J. Math. 106 (1998), 269-284. Zbl 0920.46010, MR 1656905 |
| Reference:
|
[4] Cepedello Boiso M.: On regularization in superreflexive Banach spaces by infimal convolution formulas.Studia Math. 129 (1998), 3 265-284. Zbl 0918.46014, MR 1609659 |
| Reference:
|
[5] Hartman P.: On functions representable as a difference of convex functions.Pacific J. Math. 9 (1959), 707-713. Zbl 0093.06401, MR 0110773 |
| Reference:
|
[6] Kopecká E., Malý J.: Remarks on delta-convex functions.Comment. Math. Univ. Carolinae 31.3 (1990), 501-510. MR 1078484 |
| Reference:
|
[7] Veselý L., Zajíček L.: Delta-convex mappings between Banach spaces and applications.Dissertationes Math. (Rozprawy Mat.) 289 (1989), 52 pp. MR 1016045 |
| . |
