Geometric and Metric Characterizations of Transversality Properties

Abstract

This paper continues the study of ‘good arrangements’ of collections of sets near a point in their intersection. We clarify quantitative relations between several geometric and metric characterizations of the transversality properties of collections of sets and the corresponding regularity properties of set-valued mappings. We expose all the parameters involved in the definitions and characterizations and establish relations between them. This allows us to classify the quantitative geometric and metric characterizations of transversality and regularity, and subdivide them into two groups with complete exact equivalences between the parameters within each group and clear relations between the values of the parameters in different groups.

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Acknowledgements

The authors thank the referees for the careful reading of the manuscript and their constructive comments and suggestions. We are also grateful to editor Prof. Michel Théra for organizing the refereeing process perfectly.

The research was supported by the Australian Research Council, project DP160100854. The first author is supported by an Australian Government Research Training Program (RTP) Stipend and RTP Fee-Offset Scholarship through Federation University Australia. The third author benefited from the support of the FMJH Program PGMO and Conicyt REDES program 180032.

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Correspondence to Alexander Y. Kruger.

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Dedicated to Professor Marco Antonio López Cerdá on the occasion of his 70th birthday.

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Bui, H.T., Cuong, N.D. & Kruger, A.Y. Geometric and Metric Characterizations of Transversality Properties. Vietnam J. Math. 48, 277–297 (2020). https://doi.org/10.1007/s10013-020-00388-1

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Keywords

  • Transversality
  • Subtransversality
  • Semitransversality
  • Regularity
  • Subregularity
  • Semiregularity

Mathematics Subject Classification (2010)

  • 49J52
  • 49J53
  • 49K40
  • 90C30
  • 90C46