On the concentration of measure and the $L^1$-norm

We seek a function in an $N$-dimensional subspace of $L^1$ that attains a half (more generally, a given part) of its $L^1$-norm on a set of least possible measure. We prove that such a measure is asymptotically maximal when the space is spanned by independent standard normal variables. This answers a question of Y. Benyamini, A. Kroó, A. Pinkus.