Extremality, Stationarity and Generalized Separation of Collections of Sets

TL;DR

This paper unifies and refines the theoretical foundations of extremality, stationarity, and separation in collections of sets, providing new characterizations and clarifying relationships among key parameters.

Contribution

It offers a unifying theoretical framework for extremal principles and introduces new characterizations of extremality properties in collections of sets.

Findings

Clarified quantitative relationships between parameters

Provided new characterizations of extremality properties

Unified existing approaches into a comprehensive theory

Abstract

The core arguments used in various proofs of the extremal principle and its extensions as well as in primal and dual characterizations of approximate stationarity and transversality of collections of sets are exposed, analyzed and refined, leading to a unifying theory, encompassing all existing approaches to obtaining 'extremal' statements. For that, we examine and clarify quantitative relationships between the parameters involved in the respective definitions and statements. Some new characterizations of extremality properties are obtained.