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A method for extending the representation theorems: Caps

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1757)

Abstract

The representation theorems which we dealt with in earlier sections were for elements of a compact convex set. As noted in Section 10, any such set can be regarded as a base for a closed convex cone, so these results lead in a natural way to representation theorems for the elements of a closed convex cone which admits a compact base. It is natural to wonder whether it is possible to obtain such theorems for a more general class of cones, but there seems to be no completely satisfactory result of this nature. There are, however, two lines of approach, both due to Choquet, which are of interest. One of these involves a more general notion of measure (“conical measure”), which is outlined in [19]. The other approach involves replacing the notion of “base” by that of “cap”; this makes it possible to extend the scope of the representation theorems. This section will be devoted to the latter approach. Throughout the section, we consider only proper cones K, i.e., K ∩ (-K) = ·0×.

Keywords

  • Extreme Point
  • Convex Cone
  • Lower Semicontinuous
  • Representation Theorem
  • Convex Space

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 2001 Springer-Verlag Berlin Heidelberg

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(2001). A method for extending the representation theorems: Caps. In: Phelps, R.R. (eds) Lectures on Choquet’s Theorem. Lecture Notes in Mathematics, vol 1757. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48719-0_13

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  • DOI: https://doi.org/10.1007/3-540-48719-0_13

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41834-4

  • Online ISBN: 978-3-540-48719-7

  • eBook Packages: Springer Book Archive