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2009 | 29 | 1 | 7-17
Tytuł artykułu

How to define "convex functions" on differentiable manifolds

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EN
Abstrakty
EN
In the paper a class of families 𝓕(M) of functions defined on differentiable manifolds M with the following properties:
$1_{𝓕}$. if M is a linear manifold, then 𝓕(M) contains convex functions,
$2_{𝓕}$. 𝓕(·) is invariant under diffeomorphisms,
$3_{𝓕}$. each f ∈ 𝓕(M) is differentiable on a dense $G_{δ}$-set,
is investigated.
Twórcy
  • Institute of Mathematics of the Polish Academy of Sciences, Śniadeckich 8, 00-956 Warszawa, P.O. Box 21, Poland
Bibliografia
  • [1] E. Asplund, Farthest points in reflexive locally uniformly rotund Banach spaces, Israel Jour. Math. 4 (1966), 213-216.
  • [2] E. Asplund, Fréchet differentiability of convex functions, Acta Math. 121 (1968), 31-47.
  • [3] S. Lang, Introduction to differentiable manifolds, Interscience Publishers (division of John Wiley & Sons) New York, London, 1962.
  • [4] S. Mazur, Über konvexe Mengen in linearen normierten Räumen, Stud. Math. 4 (1933), 70-84.
  • [5] E. Michael, Local properties of topological spaces, Duke Math. Jour. 21 (1954), 163-174.
  • [6] R.R. Phelps, Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics, Springer-Verlag, 1364 (1989).
  • [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.
  • [8] S. Rolewicz, On α(·)-monotone multifunction and differentiability of γ-paraconvex functions, Stud. Math. 133 (1999), 29-37.
  • 9[] S. Rolewicz, On α(·)-paraconvex and strongly α(·)-paraconvex functions, Control and Cybernetics 29 (2000), 367-377.
  • [10] S. Rolewicz, On the coincidence of some subdifferentials in the class of α(·)-paraconvex functions, Optimization 50 (2001), 353-360.
  • [11] S. Rolewicz, On uniformly approximate convex and strongly α(·)-paraconvex functions, Control and Cybernetics 30 (2001), 323-330.
  • [12] S. Rolewicz, α(·)-monotone multifunctions and differentiability of strongly α(·)-paraconvex functions, Control and Cybernetics 31 (2002), 601-619.
  • [13] S. Rolewicz, On differentiability of strongly α(·)-paraconvex functions in non-separable Asplund spaces, Studia Math. 167 (2005), 235-244.
  • [14] S. Rolewicz, Paraconvex analysis, Control and Cybernetics 34 (2005), 951-965.
  • [15] S. Rolewicz, An extension of Mazur Theorem about Gateaux differentiability, Studia Math. 172 (2006), 243-248.
  • [16] S. Rolewicz, Paraconvex Analysis on $C^{1,u}_E$-manifolds, Optimization 56 (2007), 49-60.
  • [17] L. Zajicěk, Differentiability of approximately convex, semiconcave and strongly paraconvex functions, Jour. Convex Analysis 15 (2008), 1-15.
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Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1101
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