Some notes on higher order concentration of measure

TL;DR

This paper surveys higher order concentration of measure phenomena, unifying various results for non-Lipschitz functions with bounded derivatives across different measures and geometric settings.

Contribution

It provides a comprehensive framework for higher order concentration results, extending previous work to new measures and geometric contexts, and discusses open questions.

Findings

Unified framework for higher order concentration results

Application to measures satisfying functional inequalities

Discussion of open problems in discrete settings

Abstract

This survey-type paper provides a common framework for a larger number of higher order concentration results (i.\,e., concentration results for non-Lipschitz functions which have bounded derivatives of higher order) in the spirit of Bobkov--G\"otze--Sambale (2019). Situations inlude measures satisfying various functional inequalities (log-Sobolev, Poincar\'{e}, $\mathrm{LS}_q$), uniform and cone measures on spheres with respect to the Euclidean as well as $\ell_p^n$-norms, Stiefel and Grassmann manifolds as well as discrete situations. In particular in the latter case, some open questions and remarks are stated.