Concentration between Lévy’s inequality and the Poincaré inequality for log-concave densities

Given a suitably normalized random vector [Formula: see text], we observe that the function [Formula: see text], defined for [Formula: see text], admits surprisingly strong concentration far surpassing what is expected on account of Lévy’s isoperimetric inequality. Among the measures to which the above holds are all log-concave measures, for which a solution of the similar problem concerning the third marginal moments [Formula: see text] would imply the hyperplane conjecture.