# How to define "convex functions" on differentiable manifolds

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2009)

- Volume: 29, Issue: 1, page 7-17
- ISSN: 1509-9407

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topStefan Rolewicz. "How to define "convex functions" on differentiable manifolds." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 29.1 (2009): 7-17. <http://eudml.org/doc/271159>.

@article{StefanRolewicz2009,

abstract = {In the paper a class of families (M) of functions defined on differentiable manifolds M with the following properties:
$1_\{\}$. if M is a linear manifold, then (M) contains convex functions,
$2_\{\}$. (·) is invariant under diffeomorphisms,
$3_\{\}$. each f ∈ (M) is differentiable on a dense $G_\{δ\}$-set,
is investigated.},

author = {Stefan Rolewicz},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {Fréchet differetiability; Gateaux differentiability; locally strongly paraconvex functions; $C^\{1,u\}$-manifolds; Fréchet differentiability; Gâteaux differentiability},

language = {eng},

number = {1},

pages = {7-17},

title = {How to define "convex functions" on differentiable manifolds},

url = {http://eudml.org/doc/271159},

volume = {29},

year = {2009},

}

TY - JOUR

AU - Stefan Rolewicz

TI - How to define "convex functions" on differentiable manifolds

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 2009

VL - 29

IS - 1

SP - 7

EP - 17

AB - In the paper a class of families (M) of functions defined on differentiable manifolds M with the following properties:
$1_{}$. if M is a linear manifold, then (M) contains convex functions,
$2_{}$. (·) is invariant under diffeomorphisms,
$3_{}$. each f ∈ (M) is differentiable on a dense $G_{δ}$-set,
is investigated.

LA - eng

KW - Fréchet differetiability; Gateaux differentiability; locally strongly paraconvex functions; $C^{1,u}$-manifolds; Fréchet differentiability; Gâteaux differentiability

UR - http://eudml.org/doc/271159

ER -

## References

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- 9[] S. Rolewicz, On α(·)-paraconvex and strongly α(·)-paraconvex functions, Control and Cybernetics 29 (2000), 367-377. Zbl1030.90092
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- [12] S. Rolewicz, α(·)-monotone multifunctions and differentiability of strongly α(·)-paraconvex functions, Control and Cybernetics 31 (2002), 601-619. Zbl1125.49304
- [13] S. Rolewicz, On differentiability of strongly α(·)-paraconvex functions in non-separable Asplund spaces, Studia Math. 167 (2005), 235-244. Zbl1074.46050
- [14] S. Rolewicz, Paraconvex analysis, Control and Cybernetics 34 (2005), 951-965. Zbl1167.49306
- [15] S. Rolewicz, An extension of Mazur Theorem about Gateaux differentiability, Studia Math. 172 (2006), 243-248. Zbl1106.46026
- [16] S. Rolewicz, Paraconvex Analysis on ${C}_{E}^{1,u}$-manifolds, Optimization 56 (2007), 49-60. Zbl1124.58006
- [17] L. Zajicěk, Differentiability of approximately convex, semiconcave and strongly paraconvex functions, Jour. Convex Analysis 15 (2008), 1-15. Zbl1140.46021

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