Necessary Conditions for Non-Intersection of Collections of Sets

TL;DR

This paper investigates fundamental non-intersection conditions of finite set collections, deriving new primal and dual necessary conditions in Banach/Asplund spaces, with applications to alternating projections convergence.

Contribution

It introduces novel primal and dual necessary conditions for non-intersection properties in Banach/Asplund spaces, advancing the theoretical understanding of extremality and stationarity.

Findings

New primal (slope) necessary conditions established

Dual (generalized separation) conditions derived

Applications to convergence analysis of alternating projections

Abstract

This paper continues studies of non-intersection properties of finite collections of sets initiated 40 years ago by the extremal principle. We study elementary non-intersection properties of collections of sets, making the core of the conventional definitions of extremality and stationarity. In the setting of general Banach/Asplund spaces, we establish new primal (slope) and dual (generalized separation) necessary conditions for these non-intersection properties. The results are applied to convergence analysis of alternating projections.