# About intrinsic transversality of pairs of sets

## TL;DR

This paper extends the concept of intrinsic transversality for pairs of sets to infinite-dimensional spaces, providing new characterizations and focusing on convergence of alternating projections in optimization.

## Contribution

It introduces infinite-dimensional extensions of intrinsic transversality and establishes new characterizations involving limiting objects, especially for convex sets.

## Key findings

- Characterizations of intrinsic transversality in infinite dimensions
- New limiting objects for pairs of sets
- Enhanced understanding of convergence in convex cases

## Abstract

The article continues the study of the 'regular' arrangement of a collection of sets near a point in their intersection. Such regular intersection or, in other words, transversality properties are crucial for the validity of qualification conditions in optimisation as well as subdifferential, normal cone and coderivative calculus, and convergence analysis of computational algorithms. One of the main motivations for the development of the transversality theory of collections of sets comes from the convergence analysis of alternating projections for solving feasibility problems. This article targets infinite dimensional extensions of the intrinsic transversality property introduced recently by Drusvyatskiy, Ioffe and Lewis as a sufficient condition for local linear convergence of alternating projections. Several characterisations of this property are established involving new limiting objects defined for pairs of sets. Special attention is given to the convex case.