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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 onaccount of L\'evy's isoperimetric inequality. Among the measures to which theabove holds are all log-concave measures, for which a solution of the similarproblem concerning the third marginal moments $\theta\mapsto\mathbb{E} (X\cdot\theta)^3$ would imply the hyperplane conjecture.

Concentration between L\'evy's inequality and the Poincar\'e inequality for log-concave densities