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It is known that the complex Grassmannian of $k$-dimensional subspaces can beidentified with the set of projection matrices of rank $k$. It is alsoclassically known that the convex hull of this set is the set of Hermitianmatrices with eigenvalues between $0$ and $1$ and summing to $k$. We give a newproof of this fact. We also give an existence theorem for a certaincombinatorial class of hypermatrices by a similar argument. This existencetheorem can be rewritten into an existence theorem for a uniform weightedhypergraph with given weighted degree sequence.

Convex Hulls of Grassmannians and Combinatorics of Symmetric Hypermatrices