What is the maximum volume of $N$-D slice of an $M$-D hypercube?

Solution 1:

$S(M,M-1)=\sqrt{2}$ for all $M\ge1$; this is a theorem of Keith Ball from 1986.

$S(4,2)$ is the smallest term not specified by this result; it is at least $2$, by taking the plane $x_1=x_2, x_3=x_4$, which forms a $\sqrt{2}\times\sqrt{2}$ square. I suspect this is optimal, but I don't see an easy way to prove it.

Solution 2:

Just to put into words, without proof, the idea that has been floating around here.
Let $M=aN+b$.
Split the $M$ dimensions into $b$ groups of $a+1$ and $N-b$ groups of $a$.
Take the diagonal in each group, of length $\sqrt a$ or $\sqrt{a+1}$.
The product of all these diagonals is $$\sqrt{a^{N-b}(a+1)^b}$$