Extension Complexity and Realization Spaces of Hypersimplices



The (nk)-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.


Hypersimplices Extension complexity Nonnegative rank Rectangle covering number Realization spaces 

Mathematics Subject Classification

90C57 52B12 15A23 

Supplementary material

454_2017_9925_MOESM1_ESM.py (5 kb)
Supplementary material 1 (py 4 KB)
454_2017_9925_MOESM2_ESM.py (4 kb)
Supplementary material 2 (py 3 KB)


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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Fachbereich Mathematik und InformatikFreie Universität BerlinBerlinGermany
  2. 2.Sorbonne Universités, Université Pierre et Marie Curie (Paris 6), Institut de Mathématiques de Jussieu – Paris Rive Gauche (UMR 7586)ParisFrance
  3. 3.Institut für MathematikGoethe-Universität FrankfurtFrankfurt am MainGermany

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