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Abstract

Let D be the unit disc in the complex plane ℂ. Then for every complex linear subspace H in $ℂ^n$ of codimension 1, $vol_{2n-2}(D^{n-1}) ≤ vol_{2n-2}(H ∩ D^{n}) ≤ 2vol_{2n-2}(D^{n-1})$. The lower bound is attained if and only if H is orthogonal to the versor $e_{j}$ 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 $e_{j} + σe_{k}$ for some 1 ≤ j < k ≤ n and some σ ∈ ℂ with |σ| = 1. We identify $ℂ^n$ with $ℝ^{2n}$; by $vol_{k}(·)$ we denote the usual k-dimensional volume in $ℝ^{2n}$. The result is a complex counterpart of Ball's [B1] result for cube slicing.