Given 4 variables $A$, $B$, $C$, and $D$ that are each $\pm 1$, we can build the 8 coordinates $(AB,AC,AD,BC,BD,CD)$, and then take the convex hull of these points. (There are only 8 coordinates, not 16, because flipping the sign on each variable gives the same results.) This feels -- to me -- like a reasonably natural construction, and I would like to know more about what the resulting polytope looks like. It's 6-polytope with 8 vertices. I can't find any reference to such a polytope, though, which is somewhat disappointing.
I suspect (although I'm not sure?) that the convex hull is defined entirely by the hyperplanes that pass through three related coordinates (e.g. AD, BD, and AB) and are orthogonal to the other three coordinates. But I don't know this for sure. If this is true then it would mean the polytope has 8 facets.
Can anyone tell more about this shape?
I've now learned that this is commonly referred to as the "CUT4" polytope. The "CUT4" formula typically has $(0,1)$-coordinates and my description above has $\pm 1$ coordinates, but otherwise they're easily equivalent. As an example reference for anyone interested, https://link.springer.com/article/10.1007/BF02592023 . All of the facets are of the form I describe (they only depend on 3 of the "underlying" 4 variables, and orthogonal to the other directions). Thus there are 16 facets total. Combinatorially, this is equivalent to the unique Cyclic polytope of 6 dimensions and 8 vertices, as defined as https://en.wikipedia.org/wiki/Cyclic_polytope .