Normal property, Jameson property, CHIP and linear regularity for an infinite system of convex sets in Banach spaces

Abstract

In this paper, we study different kinds of normal properties for an infinite system of arbitrarily many convex sets in a Banach space and provide the dual characterization for the normal property in terms of the extended Jameson property for arbitrarily many weak-closed convex cones in the dual space. Then, we use the normal property and the extended Jameson property to study CHIP, strong CHIP and linear regularity for the infinite case of arbitrarily many convex sets and establish equivalent relationship among these properties. In particular, we extend main results by Bakan et al. [Trans. Am. Math. Soc. 357;2005:3831–3863] on these concepts for finite system of convex sets in a Hilbert space to the infinite case of arbitrarily many convex sets in Banach space setting.

Keywords:

• Normal property
• Jameson property
• CHIP
• linear regularity

Acknowledgements

The authors are indebted to two anonymous referees for their careful readings and valuable comments and suggestions which help us improve the presentation.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundations of P.R. China [grant number 11401518]; the Fok Ying Tung Education Foundation [grant number 151101]; the Scientific Research Foundation from Education Department of Yunnan Province [grant number 2015Z009].