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6 Realizability

In this section problems are discussed which bridge the gap from combinatorial descriptions of polytopes to geometrical descriptions, i.e., it deals with questions of the following kind: given combinatorial data, does there exist a polytope which "realizes" this data? E.g., given a 0/1-matrix is this the matrix of vertex-facet incidences of a polytope? The problems of computing combinatorial from geometrical data is discussed in Section 2.

The problems listed in this section are among the first ones asked in (modern) polytope theory, going back to the work of Steinitz and Radermacher in the 1930's [61].

  • 30 Steinitz Problem
  • 31 Simplicial Steinitz Problem