Structure of the convex hull of $n$-dimensional 0/1 vectors with exactly $k$ 1s.

Solution 1:

The support vectors are the vertices of the $n$-cube $[0,1]^n$. If you orient that cube with the origin at the bottom and $(1,1,...,1)$ vertically above it. All the vertices with $\sum_i v_i = k$ will be at the same horizontal level., so $\Delta_{n,k}$ will be the intersection of a horizontal hyperplane with that solid hypercube. In general, this will be an $n-1$ dimensional object, except for the degenerate cases $\Delta_{n,0}$ and $\Delta_{n,n}$ (I don't know why you excluded the first but included the second.

The intersections are just simplicies in the hyperplane.