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Area of cross-section (at midpoint perpendicular to longest diagonal) in the unit cube of di...
For $n$ even, I get
$$\frac{\sqrt{n}}{(n-1)!} \left\langle \begin{matrix} n-1 \\ n/2-1 \end{matrix} \right\rangle.$$
Here $\left\langle \begin{matrix} a \\ b \end{matrix} \right\rangle$ is the Euleria …