Discrete & Computational Geometry

, Volume 59, Issue 3, pp 621–642 | Cite as

Extension Complexity and Realization Spaces of Hypersimplices

  • Francesco Grande
  • Arnau Padrol
  • Raman Sanyal


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 



We would like to thank Stefan Weltge, for help with the computation of rectangle covering numbers and Günter Ziegler for insightful discussions regarding realizations of hypersimplices. We are indebted to Francisco Santos for extensive discussions regarding realization spaces of hypersimplices. The coordinates for the example in Proposition 5.3 are due to him. Finally, we would like to thank the anonymous reviewers for their careful reading and useful suggestions; in particular, for pointing out the argument from [5, Lem. 3.3] which simplified and strengthened Proposition 3.4. F. Grande was supported by DFG within the research training group “Methods for Discrete Structures” (GRK1408). A. Padrol and R. Sanyal were supported by the DFG Collaborative Research Center SFB/TR 109 “Discretization in Geometry and Dynamics”. A. Padrol was also supported by the program PEPS Jeunes Chercheur-e-s of the INSMI (CNRS).

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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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