Concentration between L\'evy's inequality and the Poincar\'e 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 L\'evy'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.