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Boundaries and generation of convex sets

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Abstract

We introduce a notion which is intermediate between that of taking thew*-closed convex hull of a set and taking the norm closed convex hull of this set. This notion helps to streamline the proof (given in [FLP]) of the famous result of James in the separable case. More importantly, it leads to stronger results in the same direction. For example:

  1. 1.

    AssumeX is separable and non-reflexive and its unit sphere is covered by a sequence of balls\(\left\{ {C_i } \right\}_{i = 1}^\infty \) of radiusa<1. Then for every sequence of positive numbers\(\left\{ {\varepsilon _i } \right\}_{i = 1}^\infty \) tending to 0 there is anf εX*, such that ‖f‖ = 1 andf (x)≤1 −ε i , wheneverx εC i ,i=1,2,…

  2. 2.

    AssumeX is separable and non-reflexive and letT:Y →X* be a bounded linear non-surjective operator. Then there is anf εX* which does not attain its norm onB X such thatf ∉T(Y).

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Authors and Affiliations

  1. Department of Mathematics, Ben-Gurion University of the Negev, 84105, Beer-Sheva, Israel

    Vladimir P. Fonf

  2. Institute of Mathematics, The Hebrew University of Jerusalem Givat Ram, 91904, Jerusalem, Israel

    Joram Lindenstrauss

Authors
  1. Vladimir P. Fonf
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  2. Joram Lindenstrauss
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Correspondence to Vladimir P. Fonf.

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Fonf, V.P., Lindenstrauss, J. Boundaries and generation of convex sets. Isr. J. Math. 136, 157–172 (2003). https://doi.org/10.1007/BF02807196

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  • DOI: https://doi.org/10.1007/BF02807196

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