On Mazur rotations problem and its multidimensional versions

The article is a survey related to classical unsolved problem in Banach space theory, appearing Banach’s famous book 1932, and known as the Mazur rotations problem. Although seems very difficult rather abstract, its study sheds new light on importance of norm symmetries space, demonstrating sometimes unexpected connections with renorming theory differentiability functional analysis, topological group representations, area amenability, Fraïssé Ramsey led development concepts interest independent This focuses results that have been published after 2000, stressing two lines research which were developed last 10 years. first one approximate versions various aspects, most specifically case Lebesgue spaces $$L_p$$ . second concerns recent developments multidimensional formulations associated results. Some are also included.