Separation of Convex Sets via Barrier Cones

Abstract

A closed convex subset of a normed linear space is said to have the strong separation property if it can be strongly separated from every other disjoint, closed, and convex set by a closed hyperplane. In this paper, we give some results on the separation of convex sets noticing the role of barrier cones, develop some characterizations of subsets having the strong separation property, and apply them to consider a class of convex optimization problems.

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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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Dedicated to Professor Hoang Tuy

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Phung, H.T. Separation of Convex Sets via Barrier Cones. Acta Math Vietnam 45, 345–363 (2020). https://doi.org/10.1007/s40306-020-00367-1

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Keywords

  • Convex set
  • Separation theorem
  • Barrier cone
  • Recession cone
  • Set having the strong separation property

Mathematics Subject Classification (2010)

  • 46A55
  • 46B20
  • 52A05