# On inverses of $\delta $-convex mappings

Commentationes Mathematicae Universitatis Carolinae (2001)

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

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topDuda, 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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