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New representations of sets of lower bounds and of sets with supremum in Archimedean order-unit vector spaces

Author

Listed:
  • Emil Ernst
  • Alberto Zaffaroni

Abstract

Sets of lower bounds (also known as lower cuts), and sets with supremum, are characterized in the setting of Archimedean order-unit vector spaces (typically a Banach space ordered by means of a closed cone with non-empty interior). In this framework, our study proves that a set admits a supremum if and only if all the members of a newly defined family of supporting hyperplanes pass through a same point. This result is used to prove our second result. It states that a set is a lower cut if and only if it is bounded from above, downward, and contains the existing supremums of any of its subsets. As a consequence, we prove the following hidden convexity result: any set fulfilling the three above-mentioned conditions is necessarily a convex set.

Suggested Citation

  • Emil Ernst & Alberto Zaffaroni, 2018. "New representations of sets of lower bounds and of sets with supremum in Archimedean order-unit vector spaces," Department of Economics 0134, University of Modena and Reggio E., Faculty of Economics "Marco Biagi".
    • Handle: RePEc:mod:depeco:0134
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