  The (n,k)-hypersimplex is the convex hull of all 0/1-vectors of length n with
coordinate sum k. We explicitly determine the extension complexity of all
hypersimplices as well as of certain classes of combinatorial hypersimplices.
To that end, we investigate the projective realization spaces of hypersimplices
and their (refined) rectangle covering numbers. Our proofs combine ideas from
geometry and combinatorics and are partly computer assisted.
