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Characterisation of drop and weak drop properties for closed bounded convex sets

  • J.R. Giles (a1) and Denka N. Kutzarova (a2)
Abstract

Modifying the concept underlying Daneš' drop theorem, Rolewicz introduced the notion of the drop property of a norm which was later generalised to the weak drop property of a norm. Kutzarova extended the discussion to consider the drop property for closed bounded convex sets. Here we characterise the drop and weak drop properties for such sets by upper semi-continuous and compact valued subdifferential mappings.

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References
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[1]Bishop, E. and Phelps, R.R., ‘The support functionals of a convex set’, Proc. Symp. in Pure Maths VII, Convexity. Amer. Math. Soc. (1963), 2735.
[2]Daneš, J., ‘A geometric theorem useful in non-linear functional analysis’, Boll. Un. Mat. Ital. 6 (1972), 369372.
[3]Daneš, J., ‘Equivalence of some geometric and related results of nonlinear functional analysis’, Comm. Math. Univ. Carolinae 26 (1985), 443454.
[4]Giles, J.R., Convex analysis with application in differentiation of convex functions: Pitman Research Notes in Mathematics 58, 1982.
[5]Giles, J.R., Gregory, D.A. and Sims, Brailey, ‘Geometrical implications of upper semi-continuity of the duality mapping on a Banach space’, Pac. J. Math. 79 (1978), 99109.
[6]Giles, J.R., Sims, Brailey and Yorke, A.C., ‘On the drop and weak drop properties for a Banach space’, Bull. Austral. Math. Soc. 41 (1990), 503507.
[7]James, R.C., ‘Weakly compact sets’, Trans. Amer. Math. Soc. 113 (1964), 129140.
[8]Kutzarova, Denka N., ‘On the drop property of convex sets in Banach spaces’, Constructive theory of functions 1987, (Sofia, 1988), 283287.
[9]Kutzarova, D.N. and Rolewicz, S., ‘On drop property of convex sets’, (preprint).
[10]Phelps, R.R., Convex functions, monotone operators and differentiability. Lecture Notes in Mathematics 1364 (Springer-Verlag, 1989).
[11]Rolewicz, S., ‘On drop property’, Studia Math. 85 (1987), 2735.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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