TITLE: Area of cross-section (at midpoint perpendicular to longest diagonal) in the unit cube of dimension N 
QUESTION [8 upvotes]: Take a unit cube (of side 1) in N dimensions. Construct the cross-section at the midpoint of the longest diagonal. What is the area of this N-1 dimensional region? I can compute this, but it would be nice to have a reference and a formula to cross-check.

REPLY [7 votes]: This is a very old problem and there is a classical analytic approach to it. You can express the volume of sections of a convex body in terms of the Fourier transform of powers of the Minkowski functional. 
Let $Q_n=[-1/2,1/2]^n$ be the unit cube in $\mathbb R^n$ and let $[\xi^\perp]$ denote the hyperplane orthogonal to the vector $\xi\in S^{n-1}$. Then 
$$Vol_{n-1}(Q_n\cap [\xi^\perp])=\frac{1}{\pi}\int_{-\infty}^{\infty}
\prod\limits_{k=1}^n\frac{\sin(r \xi_k)}{r\xi_k}dr.$$
In our case $\xi=\xi_*=n^{-1/2}(1,1,\dots,1)$ so the integral becomes
$$Vol_{n-1}(Q_n\cap [\xi_*^\perp])=\frac{1}{\pi}\int_{-\infty}^{\infty}
\left(\frac{\sin(r/\sqrt n)}{r/\sqrt n}\right)^ndr.$$
In fact, the latter formula was already known to Laplace! The integral tends to $\sqrt {6/\pi}$, as $n\to\infty$ (e.g. by Laplace's method).  This can be also justified via the probabilistic interpretation suggested by Michael Lugo.
The Fourier analytic approach to sections of convex bodies is nicely presented in The Interface between Convex Geometry and Harmonic Analysis by A. Koldobsky and V. Yaskin. The derivation of the formula for volumes is available here.
EDIT (15.01.2011). In fact both integrals can be calculated explicitly. The sinc integrals were studied by Borwein & Borwein (see also Multi-Variable sinc Integrals and the Volumes of Polyhedra).
For any $n\in \mathbb N$, $n>1$, we have
$$\frac{1}{\pi}\int_{-\infty}^{\infty}
\left(\frac{\sin(r/\sqrt n)}{r/\sqrt n}\right)^ndr =\frac{\sqrt n}{2^{n-1}(n-1)!}\sum_{k=0}^{n/2}(-1)^k{n \choose k}(n-2k)^{n-1}$$