Non-central sections of the $l_1$-ball

Abstract

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