"A body like the cube, which is bounded by a finite number of flat facets, is called a polytope. Among symmetric polytopes, the cube has the fewest possible facets, namely $2n$."
I am looking for rigorous proof and also intuition behind why this must be true. I'm thinking of doing something by contradiction, i.e. assuming less than $2n$ facets for a symmetric polytope and then going ahead from there?
I don't have much to add here in terms of how I approached this, because this isn't a problem - it is something I came across while reading some lecture notes! I could not find much on this online, and I'd appreciate any help.
Edit: It seems the author is talking about centrally symmetric polytopes, i.e. $x\in K$ whenever $-x\in K$.
By symmetry, any face should have an opposite face. So, your polytope should have at least two opposite faces. But the region bounded between two opposite hyperplanes is not bounded, you still have $n-1$ degrees of freedom inside the "slice" of $\mathbb{R}^n$. You choose another pair of independent (opposite) hyperplanes. These bound your region in an independent direction, but you still have $n-2$ directions along which it is unbounded. You have to continue until you have at least $2n$ faces to get a bounded region. Since a cube has exactly $2n$ faces, it is the polytope with the minimum possible number of them.