On Mazur rotations problem and its multidimensional versions

TL;DR

This survey reviews recent advances in the Mazur rotations problem, exploring approximate solutions in Lebesgue spaces and multidimensional versions, highlighting connections with various areas of functional analysis and related fields.

Contribution

It provides a comprehensive overview of developments after 2000, including new results on approximate and multidimensional formulations of the problem.

Findings

Progress on approximate Mazur rotations in Lp spaces

Development of multidimensional versions of the problem

New results linking norm symmetries with other mathematical areas

Abstract

The article is a survey related to a classical unsolved problem in Banach space theory, appearing in Banach's famous book in 1932, and known as the Mazur rotations problem. Although the problem seems very difficult and rather abstract, its study sheds new light on the importance of norm symmetries of a Banach space, demonstrating sometimes unexpected connections with renorming theory and differentiability in functional analysis, with topological group theory and the theory of representations, with the area of amenability, with Fra\"iss\'e theory and Ramsey theory, and led to development of concepts of interest independent of Mazur problem. This survey focuses on results that have been published after 2000, stressing two lines of research which were developed in the last ten years. The first one is the study of approximate versions of Mazur rotations problem in its various aspects, most specifically in the case of the Lebesgue spaces Lp. The second one concerns recent developments of multidimensional formulations of Mazur rotations problem and associated results. Some new results are also included.