Convex Hulls of Grassmannians and Combinatorics of Symmetric Hypermatrices

TL;DR

This paper provides a new proof for the convex hull of Grassmannian projection matrices and establishes an existence theorem for certain hypermatrices and hypergraphs using combinatorial and convex geometric methods.

Contribution

It introduces a novel proof of the convex hull characterization of Grassmannians and extends the approach to hypermatrices and hypergraphs, linking convex geometry with combinatorics.

Findings

Convex hull of Grassmannian projection matrices is the set of Hermitian matrices with eigenvalues in [0,1] summing to k.

Existence theorem for a class of hypermatrices using convex geometric arguments.

Correspondence between hypermatrices and weighted hypergraphs with specified degree sequences.

Abstract

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.