Polydisc slicing in ${\displaystyle {ℂ}^n}$

Let D be the unit disc in the complex plane ℂ. Then for every complex linear subspace H in ${\displaystyle {ℂ}^n}$ of codimension 1, ${\displaystyle vo{l}_{2n-2}\left(D^{n-1}\right)\le vo{l}_{2n-2}(H\cap D^n)\le 2vo{l}_{2n-2}\left(D^{n-1}\right)}$. The lower bound is attained if and only if H is orthogonal to the versor ${\displaystyle e_j}$ of the jth coordinate axis for some j = 1,...,n; the upper bound is attained if and only if H is orthogonal to a vector ${\displaystyle e_j+\sigma e_k}$ for some 1 ≤ j < k ≤ n and some σ ∈ ℂ with |σ| = 1. We identify ${\displaystyle {ℂ}^n}$ with ${\displaystyle {ℝ}^{2n}}$; by ${\displaystyle vo{l}_k(·)}$ we denote the usual k-dimensional volume in ${\displaystyle {ℝ}^{2n}}$. The result is a complex counterpart of Ball's [B1] result for cube slicing.