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The Existence of Unbounded Closed Convex Sets with Trivial Recession Cone in Normed Spaces

Abstract

It’s well known that a closed convex set in a finite-dimensional normed space is unbounded if and only if it has a nonzero recession direction. In this work, we shall prove that in every infinite-dimensional normed space there exists an unbounded closed convex set whose recession cone consists of the zero vector alone.

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Acknowledgments

The author would like to thank the referees for their helpful comments and valuable suggestions.

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Correspondence to Huynh The Phung.

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Phung, H.T. The Existence of Unbounded Closed Convex Sets with Trivial Recession Cone in Normed Spaces. Acta Math Vietnam 41, 277–282 (2016). https://doi.org/10.1007/s40306-015-0131-2

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  • DOI: https://doi.org/10.1007/s40306-015-0131-2

Keywords

  • Recession cone
  • Bounded and linearly bounded
  • Convex set
  • Infinite-dimensional normed space

Mathematics Subject Classification (2010)

  • 52A05
  • 52A07
  • 46B99
  • 46N10