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

Abstract

Given a suitably normalized $X\in {\mathbb{R}}^n$ we observe that the function $\theta \mapsto \mathbb{E}|X\cdot \theta |$, defined for $\theta \in S^{n-1}$, 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 $\theta \mapsto \mathbb{E}(X\cdot \theta {)}^3$ would imply the hyperplane conjecture.

arXiv commentExtended version including an appendix