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Demyanov Difference in Infinite-Dimensional Spaces

  • Jerzy Grzybowski
  • Diethard Pallaschke
  • Ryszard Urbański
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 87)

Abstract

In this paper we generalize the Demyanov difference to the case of real Hausdorff topological vector spaces. We prove some classical properties of the Demyanov difference. In the proofs we use a new technique which is based on the properties given in Lemma 1. Due to its importance it will be called the preparation lemma. Moreover, we give connections between Minkowski subtraction and the union of upper differences. We show that in the case of normed spaces the Demyanov difference coincides with classical definitions of Demyanov subtraction.

Keywords

Minkowski subtraction Demyanov difference Pairs of closed bounded convex sets 

References

  1. [1]
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)MATHGoogle Scholar
  2. [2]
    Demyanov, V.F., Rubinov, A.M.: Quasidifferential Calculus. Springer and Optimization Software Inc., New York (1986)CrossRefMATHGoogle Scholar
  3. [3]
    Hukuhara, M.: Intégration des applications mesurables dont la valeur est un compact convexe (French). Funkcial. Ekvac. 10, 205–223 (1967)MathSciNetMATHGoogle Scholar
  4. [4]
    Husain, T., Tweddle, I.: On the extreme points of the sum of two compact convex sets. Math. Ann. 188, 113–122 (1970)MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    Klee, V.: Extremal structure of convex sets II. Math. Zeitschr. 69, 90–104 (1958)MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    Pallaschke, D., Przybycień, H., Urbański, R.: On partialy ordered semigroups. J. Set Valued Anal. 16, 257–265 (2007)CrossRefGoogle Scholar
  7. [7]
    Pallaschke, D., Urbański, R.: Pairs of Compact Convex Sets: Fractional Arithmetic with Convex Sets, Mathematics and Its Applications. Kluwer Academic Publisher, Dortrecht (2002)CrossRefGoogle Scholar
  8. [8]
    Pallaschke, D., Urbański, R.: On the separation and order law of cancellation for bounded sets. Optimization 51, 487–496 (2002)MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    Pinsker A.G.: The space of convex sets of a locally convex space. Trudy Leningrad Eng. Econ. Inst. 63, 13–17 (1966)MathSciNetGoogle Scholar
  10. [10]
    Pontryagin L.S.: On linear differential games II (Russian). Dokl. Acad. Nauk SSSR 175, 764–766 (1967)Google Scholar
  11. [11]
    Rubinov A.M., Akhundov, I.S.: Differences of compact sets in the sense of Demyanov and its application to non-smooth-analysis. Optimization 23, 179–189 (1992)MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    Urbański, R.: A generalization of the Minkowski–Rådström–Hörmander theorem. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24, 709–715 (1976)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Jerzy Grzybowski
    • 1
  • Diethard Pallaschke
    • 2
  • Ryszard Urbański
    • 1
  1. 1.Faculty of Mathematics and Computer ScienceAdam Mickiewicz UniversityPoznańPoland
  2. 2.Institute of Operations ResearchUniversity of Karlsruhe (KIT)KarlsruheGermany

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