Generalized Separation of Collections of Sets

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.

21 pages, Optimization (2025)

References in corpus (6)

• About uniform regularity of collections of sets
• Extremality, Stationarity and Generalized Separation of Collections of Sets
• Transversality Properties: Primal Sufficient Conditions
• About Extensions of the Extremal Principle
• Extremality of families of sets
• Sequential Extremal Principle and Necessary Conditions for Minimizing Sequences