Consider a set in $R^n$ defined as the intersection of the unit hypercube $[0,1]^n$ with a hyperplane defined by $\sum x_i = k$. Assume $k \in (0,n)$ so the intersection has positive volume.
Can we characterize the resulting (n-1)-polytope? In particular, does there exist a condition where the polytope is a simplex? I am looking at an application of Stokes Theorem, and trying to figure out if its 2-faces are guaranteed symmetrical.
Edit: (New material below) Thanks for the comments. I looked up hypersimplex in much more detail.
Further reading and thinking has definitely clarified that the 2-faces are not guaranteed triangular, but are they perhaps symmetrical? This arXiv article seems to indicate that each vertex will have the same number of adjacent vertices... but I can't tell if integer k's are implicitly assumed. If correct for arbitrary k, then does this give symmetry of the 2-faces? Say, all equal area/circumference shapes, depending on k?
Intuitively, it seems there would be symmetry because all vertices are combinations of the same number of zeros, ones, and one k-1 element. This arXiv article seems to indicate that the polytope can be broken down into lower dimensional simplices, which also suggests some symmetry? I am having a hard time understanding the definitions in this one, though... maybe they have already solved my problem for me?