Generalized Separation of Collections of Sets

TL;DR

This paper develops a unified framework for generalized separation of collections of sets using arbitrary product norms, extending existing principles and providing dual conditions for related properties.

Contribution

It introduces a general separation theorem applicable to various product norms and details the proof structure, advancing the theoretical understanding of set separation.

Findings

Unified separation theorem for arbitrary product norms

Dual conditions for approximate stationarity and transversality

Framework unifies and extends existing separation principles

Abstract

We show that the existing generalized separation statements including the conventional extremal principle and its extensions differ {in the ways norms on product spaces are defined}. We prove a general separation statement with arbitrary product norms covering the existing results of this kind. The proof is divided into a series of claims and exposes the key steps and arguments used when proving generalized separation statements. As an application, we prove dual necessary (sufficient) conditions for an abstract product norm extension of the approximate stationarity (transversality) property.