  It is known that the complex Grassmannian of $k$-dimensional subspaces can be
identified with the set of projection matrices of rank $k$. It is also
classically known that the convex hull of this set is the set of Hermitian
matrices with eigenvalues between $0$ and $1$ and summing to $k$. We give a new
proof of this fact. We also give an existence theorem for a certain
combinatorial class of hypermatrices by a similar argument. This existence
theorem can be rewritten into an existence theorem for a uniform weighted
hypergraph with given weighted degree sequence.
