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Israel Journal of Mathematics

, Volume 57, Issue 3, pp 257–271 | Cite as

Polytopal and nonpolytopal spheres an algorithmic approach

  • Jürgen Bokowski
  • Bernd Sturmfels
Article

Abstract

The convexity theory for oriented matroids, first developed by Las Vergnas [17], provides the framework for a new computational approach to the Steinitz problem [13]. We describe an algorithm which, for a given combinatorial (d − 2)-sphereS withn vertices, determines the setC d,n(S) of rankd oriented matroids withn points and face latticeS. SinceS is polytopal if and only if there is a realizableM εC d,n(S), this method together with the coordinatizability test for oriented matroids in [10] yields a decision procedure for the polytopality of a large class of spheres. As main new result we prove that there exist 431 combinatorial types of neighborly 5-polytopes with 10 vertices by establishing coordinates for 98 “doubted polytopes” in the classification of Altshuler [1]. We show that for allnk + 5 ≧8 there exist simplicialk-spheres withn vertices which are non-polytopal due to the simple fact that they fail to be matroid spheres. On the other hand, we show that the 3-sphereM 963 9 with 9 vertices in [2] is the smallest non-polytopal matroid sphere, and non-polytopal matroidk-spheres withn vertices exist for allnk + 6 ≧ 9.

Keywords

Decision Procedure Face Lattice Convex Polytopes Combinatorial Type Inequality System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Hebrew Univeristy 1987

Authors and Affiliations

  • Jürgen Bokowski
    • 1
  • Bernd Sturmfels
    • 1
  1. 1.Technische Hochschule DarmstadtFachbereich Mathematik, AG 3DarmstadtFRG

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