This is surely well known, but:

Q1. What is the $(d{-}1)$-dimensional polytope that realizes the maximum volume cross-section of a unit hypercube by a $(d{-}1)$-dimensional hyperplane?

*Answered* in the comments: Ball's theorem.

^{ (Image from MathWorld. This is not the maximum area slice [Noam Elkies].) }

I am less interested in the numerical maximum volume as I am in a description of the polytope that achieves it.

^{ (4D slicing snapshots from eusebeia.dyndns.org.) }

Q2. Is the polytope that answers Q1 is the same as that which achieves the maximum volume orthogonal projection (shadow) of the hypercube?

*Answered* by Paata Ivanisvili: No for $d \ge 3$.

Q1: As I recall the central section orthogonal to $(1,1,1,\ldots,1)$ gives a cross-section approaching $\sqrt{6/\pi}$ for large $n$, which is less than the $\sqrt 2$ of the central section orthogonal to $(1,1,0,0,0,\ldots,0)$. Certainly in dimension $3$ the hexagon is pretty but smaller (area $\sqrt{27/16} < 1.3 < 1.4 < \sqrt2$). That $\sqrt 2$ is supposed to be maximal but I don't know/remember whether this has been proved. $\endgroup$ – Noam D. Elkies Aug 20 '16 at 1:02