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Set-Valued Evenly Convex Functions: Characterizations and C-Conjugacy

Abstract

In this work we deal with set-valued functions with values in the power set of a separated locally convex space where a nontrivial pointed convex cone induces a partial order relation. A set-valued function is evenly convex if its epigraph is an evenly convex set, i.e., it is the intersection of an arbitrary family of open half-spaces. In this paper we characterize evenly convex set-valued functions as the pointwise supremum of its set-valued e-affine minorants. Moreover, a suitable conjugation pattern will be developed for these functions, as well as the counterpart of the biconjugation Fenchel-Moreau theorem.

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Acknowledgements

The author sincerely thanks anonymous referees for their careful reading and thoughtful comments. Their suggestions have significantly improved the quality of the paper.

Research partially supported by MINECO of Spain and ERDF of EU, Grant MTM2014-59179-C2-1-P.

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Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.

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Correspondence to M. D. Fajardo.

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Fajardo, M.D. Set-Valued Evenly Convex Functions: Characterizations and C-Conjugacy. Set-Valued Var. Anal (2022). https://doi.org/10.1007/s11228-021-00621-0

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  • DOI: https://doi.org/10.1007/s11228-021-00621-0

Keywords

  • Evenly convex sets
  • Set-valued functions
  • Partially ordered spaces
  • Convex conjugation
  • Fenchel-Moreau theorem

Mathematics Subject Classification (2010)

  • 49N15
  • 52A41
  • 90C25