# A d.c. ${C}^{1}$ function need not be difference of convex ${C}^{1}$ functions

Commentationes Mathematicae Universitatis Carolinae (2005)

- Volume: 46, Issue: 1, page 75-83
- ISSN: 0010-2628

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topPavlica, David. "A d.c. $C^1$ function need not be difference of convex $C^1$ functions." Commentationes Mathematicae Universitatis Carolinae 46.1 (2005): 75-83. <http://eudml.org/doc/249529>.

@article{Pavlica2005,

abstract = {In [2] a delta convex function on $\mathbb \{R\}^2$ is constructed which is strictly differentiable at $0$ but it is not representable as a difference of two convex function of this property. We improve this result by constructing a delta convex function of class $C^1(\mathbb \{R\}^2)$ which cannot be represented as a difference of two convex functions differentiable at 0. Further we give an example of a delta convex function differentiable everywhere which is not strictly differentiable at 0.},

author = {Pavlica, David},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {differentiability; delta-convex functions; diferentiability; delta-convex functions},

language = {eng},

number = {1},

pages = {75-83},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {A d.c. $C^1$ function need not be difference of convex $C^1$ functions},

url = {http://eudml.org/doc/249529},

volume = {46},

year = {2005},

}

TY - JOUR

AU - Pavlica, David

TI - A d.c. $C^1$ function need not be difference of convex $C^1$ functions

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 2005

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 46

IS - 1

SP - 75

EP - 83

AB - In [2] a delta convex function on $\mathbb {R}^2$ is constructed which is strictly differentiable at $0$ but it is not representable as a difference of two convex function of this property. We improve this result by constructing a delta convex function of class $C^1(\mathbb {R}^2)$ which cannot be represented as a difference of two convex functions differentiable at 0. Further we give an example of a delta convex function differentiable everywhere which is not strictly differentiable at 0.

LA - eng

KW - differentiability; delta-convex functions; diferentiability; delta-convex functions

UR - http://eudml.org/doc/249529

ER -

## References

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- Penot J.-P., Bougeard M.L., Approximation and decomposition properties of some classes of locally d.c. functions, Math. Programming 41 (1988), 195-227. (1988) Zbl0666.49005MR0945661
- Rockafellar R.T., Convex Analysis, Princeton University Press, Princeton (1970). (1970) Zbl0193.18401MR0274683
- Shapiro A., On functions representable as a difference of two convex functions in inequality constrained optimization, Research report University of South Africa, 1983.
- Veselý L., Zajíček L., Delta-convex mappings between Banach spaces and applications, Dissertationes Math. 289 (1989), 1-52. (1989) MR1016045

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