Israel Journal of Mathematics

, Volume 211, Issue 1, pp 239–255 | Cite as

A universality theorem for projectively unique polytopes and a conjecture of Shephard



We prove that every polytope described by algebraic coordinates is the face of a projectively unique polytope. This provides a universality property for projectively unique polytopes. Using a closely related result of Below, we construct a combinatorial type of 5-dimensional polytope that is not realizable as a subpolytope of any stacked polytope. This disproves a classical conjecture in polytope theory, first formulated by Shephard in the seventies.


Unique Point Projective Transformation Combinatorial Type Projective Basis Oriented Matroid 
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  1. [1]
    K. A. Adiprasito and G. M. Ziegler, Many projectively unique polytopes, Inventiones mathematicae 199 (2015), 581–652.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    A. Below, Complexity of triangulation, Ph.D. thesis, ETH Zürich, Zürich, CH, 2002.Google Scholar
  3. [3]
    E. Bierstone and P. D. Milman, Semianalytic and subanalytic sets, Institut des Hautes Études Scientifiques. Publications Mathématiques 67 (1988), 5–42.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    M. Dobbins, Representations of polytopes, Ph.D. thesis, Temple University, Philadelphia, PA, 2011.Google Scholar
  5. [5]
    B. Grünbaum, Convex Polytopes, 2nd ed., Graduate Texts in Mathematics, Vol. 221, Springer, New York, 2003.CrossRefMATHGoogle Scholar
  6. [6]
    G. Kalai, Polytope skeletons and paths, in Handbook of Discrete and Computational Geometry, Chapman & Hall/CRC, Boca Raton, FL, 2004, pp. 455–476.Google Scholar
  7. [7]
    G. Kalai, Open problems for convex polytopes I’d love to see solved, Talk on Workshop ”Convex Polytopes” at RIMS Kyoto, July 2012, slides available at
  8. [8]
    M. Kapovich and J. J. Millson, Moduli spaces of linkages and arrangements, in Advances in Geometry, Progress in Mathematics, Vol. 172, Birkhäuser Boston, Boston, MA, 1999, pp. 237–270.CrossRefGoogle Scholar
  9. [9]
    M. K”omhoff, On a combinatorial problem concerning subpolytopes of stack polytopes, Geometriae Dedicata 9 (1980), 73–76.MathSciNetGoogle Scholar
  10. [10]
    B. Lindström, On the realization of convex polytopes, Euler’s formula and Möbius functions, Aequationes Mathematicae 6 (1971), 235–240.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    J. Richter-Gebert, Realization Spaces of Polytopes, Lecture Notes in Mathematics, Vol. 1643, Springer, Berlin, 1996.MATHGoogle Scholar
  12. [12]
    J. Richter-Gebert, Perspectives on Projective Geometry, Springer, Heidelberg, 2011.CrossRefMATHGoogle Scholar
  13. [13]
    R. Schneider, Convex Bodies: the Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, Vol. 44, Cambridge University Press, Cambridge, 1993.CrossRefMATHGoogle Scholar
  14. [14]
    G. C. Shephard, Subpolytopes of stack polytopes, Israel Journal of Mathematics 19 (1974), 292–296.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    K. G. C. von Staudt, Beiträge zur Geometrie der Lage, no. 2, Baur und Raspe, Nürnberg, 1857.Google Scholar

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© Hebrew University of Jerusalem 2016

Authors and Affiliations

  1. 1.Institut des Hautes Études ScientifiquesBures-sur-YvetteFrance
  2. 2.Institut für MathematikFU BerlinBerlinGermany

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