Manifestations of Non Linear Roundness in Analysis, Discrete Geometry and Topology

Some forty years ago Per Enflo introduced the nonlinear notions of roundness and generalized roundness for general metric spaces in order to study (a) uniform homeomorphisms between (quasi-) Banach spaces, and (b) Hilbert’s Fifth Problem in the context of non locally compact topological groups (see [24], [25], [26], and [27]). Since then the concepts of roundness and generalized roundness have proven to be particularly useful and durable across a number of important mathematical fields such as coarse geometry, discrete geometry, functional analysis and topology. The purpose of this article is to take a retrospective look at some notable applications of versions of nonlinear roundness across such fields, to draw some hitherto unpublished connections between such results, and to highlight some very intriguing open problems.