Extension complexity and realization spaces of hypersimplices

TL;DR

This paper determines the extension complexity of hypersimplices and explores their realization spaces, combining geometric and combinatorial methods, including computer-assisted proofs.

Contribution

It provides explicit extension complexity values for all hypersimplices and analyzes their realization spaces, advancing understanding of their geometric and combinatorial properties.

Findings

Extension complexity of all hypersimplices explicitly determined

Analysis of projective realization spaces of hypersimplices

Use of computer-assisted proofs in geometric combinatorics

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.