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1993-1995 | 22 | 3 | 419-426
Tytuł artykułu

Some remarks on the space of differences of sublinear functions

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Two properties concerning the space of differences of sublinear functions D(X) for a real Banach space X are proved. First, we show that for a real separable Banach space (X,‖·‖) there exists a countable family of seminorms such that D(X) becomes a Fréchet space. For X = ℝ^n this construction yields a norm such that D(ℝ^n) becomes a Banach space. Furthermore, we show that for a real Banach space with a smooth dual every sublinear Lipschitzian function can be expressed by the Fenchel conjugate of the farthest point mapping to its subdifferential at the origin. This leads to a simple family of sublinear functions which contains an exhaustive family of upper convex approximations for any quasidifferentiable function.
Rocznik
Tom
22
Numer
3
Strony
419-426
Opis fizyczny
Daty
wydano
1994
otrzymano
1993-12-30
Twórcy
  • Institut für Statistik und Mathematische Wirtschaftstheorie, Universität Karlsruhe, Kaiserstr. 12, D-76128 Karlsruhe, Germany
  • Institut für Statistik und Mathematische Wirtschaftstheorie, Universität Karlsruhe, Kaiserstr. 12, D-76128 Karlsruhe, Germany
Bibliografia
  • [1] U. Cegrell, On the space of delta-convex functions and its dual, Bull. Math. Soc. Sci. Math. R. S. Roumanie 22 (1978), 133-139.
  • [2] V. F. Demyanov and A. M. Rubinov, Quasidifferential Calculus, Optimization Software Inc., Publications Division, New York, 1986.
  • [3] J. Diestel, Geometry of Banach Spaces-Selected Topics, Lecture Notes in Math. 485, Springer, Heidelberg, 1975.
  • [4] J. Grzybowski, Minimal pairs of compact convex sets, Arch. Math. (Basel), submitted.
  • [5] L. Hörmander, Sur la fonction d'appui des ensembles convexes dans un espace localement convexe, Ark. Mat. 3 (1954), 181-186.
  • [6] D. Pallaschke, P. Recht and R. Urbański, On locally Lipschitz quasidifferentiable functions in Banach spaces, Optimization 17 (1986), 287-295.
  • [7] D. Pallaschke, S. Scholtes and R. Urbański, On minimal pairs of compact convex sets, Bull. Polish Acad. Sci. Math. 39 (1991), 1-5.
  • [8] D. Pallaschke and R. Urbański, Some criteria for the minimality of pairs of compact convex sets, Z. Oper. Res. 37 (1993), 129-150.
  • [9] D. Pallaschke and R. Urbański, Reduction of quasidifferentials and minimal representations, Math. Programming Ser. A, to appear.
  • [10] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N.J., 1972.
  • [11] S. Rolewicz, Metric Linear Spaces, PWN, Warszawa, and Reidel, Boston, 1984.
  • [12] H. H. Schäfer, Topological Vector Spaces, Springer, New York, 1971.
  • [13] S. Scholtes, Minimal pairs of convex bodies in two dimensions, Mathematika 39 (1992), 267-273.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-zmv22z3p419bwm
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