Polydisc slicing in C-n

Let D be the unit disc in the complex plane C. Then for every complex linea r subspace H in C-n of codimension 1, vol(2n-2)(Dn-1) less than or equal to vol(2n-2)(H boolean AND D-n) less tha n or equal to 2vol(2n-2)(Dn-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 att ained if and only if H is orthogonal to a vector e(j) + sigmae(k) for some 1 less than or equal to j < k <less than or equal to> n and some sigma is a n element of C with \ sigma \ = 1. We identify C-n with R-2n; by vol(k)(.) we denote the usual k-dimensional v olume in R-2n. The result is a complex counterpart of Ball's [B1] result fo r cube slicing.