How to define "convex functions" on differentiable manifolds

Stefan Rolewicz

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

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

Abstract

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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.

How to cite

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Stefan 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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  1. [1] E. Asplund, Farthest points in reflexive locally uniformly rotund Banach spaces, Israel Jour. Math. 4 (1966), 213-216. Zbl0143.34904
  2. [2] E. Asplund, Fréchet differentiability of convex functions, Acta Math. 121 (1968), 31-47. Zbl0162.17501
  3. [3] S. Lang, Introduction to differentiable manifolds, Interscience Publishers (division of John Wiley & Sons) New York, London, 1962. Zbl0103.15101
  4. [4] S. Mazur, Über konvexe Mengen in linearen normierten Räumen, Stud. Math. 4 (1933), 70-84. Zbl59.1074.01
  5. [5] E. Michael, Local properties of topological spaces, Duke Math. Jour. 21 (1954), 163-174. 
  6. [6] R.R. Phelps, Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics, Springer-Verlag, 1364 (1989). Zbl0658.46035
  7. [7] D. Preiss and L. Zajíček, Stronger estimates of smallness of sets of Fréchet nondifferentiability of convex functions, Proc. 11-th Winter School, Suppl. Rend. Circ. Mat di Palermo, ser II, 3 (1984), 219-223. Zbl0547.46026
  8. [8] S. Rolewicz, On α(·)-monotone multifunction and differentiability of γ-paraconvex functions, Stud. Math. 133 (1999), 29-37. Zbl0920.47047
  9. 9[] S. Rolewicz, On α(·)-paraconvex and strongly α(·)-paraconvex functions, Control and Cybernetics 29 (2000), 367-377. Zbl1030.90092
  10. [10] S. Rolewicz, On the coincidence of some subdifferentials in the class of α(·)-paraconvex functions, Optimization 50 (2001), 353-360. Zbl1008.52003
  11. [11] S. Rolewicz, On uniformly approximate convex and strongly α(·)-paraconvex functions, Control and Cybernetics 30 (2001), 323-330. Zbl1038.26009
  12. [12] S. Rolewicz, α(·)-monotone multifunctions and differentiability of strongly α(·)-paraconvex functions, Control and Cybernetics 31 (2002), 601-619. Zbl1125.49304
  13. [13] S. Rolewicz, On differentiability of strongly α(·)-paraconvex functions in non-separable Asplund spaces, Studia Math. 167 (2005), 235-244. Zbl1074.46050
  14. [14] S. Rolewicz, Paraconvex analysis, Control and Cybernetics 34 (2005), 951-965. Zbl1167.49306
  15. [15] S. Rolewicz, An extension of Mazur Theorem about Gateaux differentiability, Studia Math. 172 (2006), 243-248. Zbl1106.46026
  16. [16] S. Rolewicz, Paraconvex Analysis on C E 1 , u -manifolds, Optimization 56 (2007), 49-60. Zbl1124.58006
  17. [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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