Non-central sections of the l1-ball

We determine the maximal non-central hyperplane sections of the $l_1^n$-ball if the fixed distance of the hyperplane to the origin is between $\frac{1}{\sqrt{3}}$ and $\frac{1}{\sqrt{2}}$. This adds to a result of Liu and Tkocz who considered the distance range between $\frac{1}{\sqrt{2}}$ and 1. For $n\ge 4$, the maximal sections are parallel to the $(n-1)$-dimensional coordinate planes. We also study non-central sections of the complex $l_{\infty }^2$-ball, where the formulas are more complicated than in the real case. Also, the extrema are partially different compared to the real case.