Some notes on higher order concentration of measure

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--Götze--Sambale (2019). Situations inlude measures satisfying various functional inequalities (log-Sobolev, Poincaré, ${\mathrm{LS}}_q$), 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.

generalization of various previous results plus some context and open questions