Extension complexity and realization spaces of hypersimplices

Abstract

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.

17 pages, 3 figures, 2 python scripts as part of the submission; v2: Rectangle covering number bounds improved (following suggestion of the referees), Results added on prescribability of facets of combinatorial hypersimplices, minor fixes