On inverses of δ -convex mappings

Jakub Duda

Commentationes Mathematicae Universitatis Carolinae (2001)

  • Volume: 42, Issue: 2, page 281-297
  • ISSN: 0010-2628

Abstract

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In the first part of this paper, we prove that in a sense the class of bi-Lipschitz δ -convex mappings, whose inverses are locally δ -convex, is stable under finite-dimensional δ -convex perturbations. In the second part, we construct two δ -convex mappings from ℓ 1 onto ℓ 1 , which are both bi-Lipschitz and their inverses are nowhere locally δ -convex. The second mapping, whose construction is more complicated, has an invertible strict derivative at 0 . These mappings show that for (locally) δ -convex mappings an infinite-dimensional analogue of the finite-dimensional theorem about δ -convexity of inverse mappings (proved in [7]) cannot hold in general (the case of ℓ 2 is still open) and answer three questions posed in [7].

How to cite

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Duda, Jakub. "On inverses of $\delta $-convex mappings." Commentationes Mathematicae Universitatis Carolinae 42.2 (2001): 281-297. <http://eudml.org/doc/248765>.

@article{Duda2001,
abstract = {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].},
author = {Duda, Jakub},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {delta-convex mappings; strict differentiability; normed linear spaces; delta-convex mappings; normed linear spaces; strict differentiability},
language = {eng},
number = {2},
pages = {281-297},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On inverses of $\delta $-convex mappings},
url = {http://eudml.org/doc/248765},
volume = {42},
year = {2001},
}

TY - JOUR
AU - Duda, Jakub
TI - On inverses of $\delta $-convex mappings
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2001
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 42
IS - 2
SP - 281
EP - 297
AB - 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].
LA - eng
KW - delta-convex mappings; strict differentiability; normed linear spaces; delta-convex mappings; normed linear spaces; strict differentiability
UR - http://eudml.org/doc/248765
ER -

References

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  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). (1949) MR0048059
  2. Alexandrov A.D., Surfaces represented by the differences of convex functions, Doklady Akad. Nauk SSSR (N.S.) 72 (1950), 613-616 (in Russian). (1950) MR0038092
  3. Cepedello Boiso M., Approximation of Lipschitz functions by Δ -convex functions in Banach spaces, Israel J. Math. 106 (1998), 269-284. (1998) Zbl0920.46010MR1656905
  4. Cepedello Boiso M., On regularization in superreflexive Banach spaces by infimal convolution formulas, Studia Math. 129 (1998), 3 265-284. (1998) Zbl0918.46014MR1609659
  5. Hartman P., On functions representable as a difference of convex functions, Pacific J. Math. 9 (1959), 707-713. (1959) Zbl0093.06401MR0110773
  6. Kopecká E., Malý J., Remarks on delta-convex functions, Comment. Math. Univ. Carolinae 31.3 (1990), 501-510. (1990) MR1078484
  7. Veselý L., Zajíček L., Delta-convex mappings between Banach spaces and applications, Dissertationes Math. (Rozprawy Mat.) 289 (1989), 52 pp. (1989) MR1016045

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