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

TL;DR

This paper extends the concepts of normal properties, Jamenson property, CHIP, and linear regularity from finite to infinite systems of convex sets in Banach spaces, providing dual characterizations and establishing their equivalences.

Contribution

It introduces dual characterizations of normal properties for infinite convex systems and extends finite-system results to infinite systems in Banach spaces.

Findings

Dual characterization of normal property via Jamenson property.

Equivalence among normal property, CHIP, and linear regularity for infinite systems.

Extension of finite-system results to infinite convex sets in Banach spaces.

Abstract

In this paper, we study different kinds of normal properties for 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 Jamenson property for arbitrarily many weak*-closed convex cones in the dual space. Then, we use the normal property and the extended Jamenson 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 in [3] on normal property, Jamenson property, CHIP and linear regularity for finite system of convex sets in a Hilbert space to the infinite case of arbitrarily many convex sets in Banach space setting.