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Boundary Complexes of Convex Polytopes cannot Be Characterized Locally
Author(s) -
Sturmfels Bernd
Publication year - 1987
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-35.2.314
Subject(s) - matroid , polytope , combinatorics , mathematics , realizability , polyhedron , regular polygon , convex polytope , boundary (topology) , lattice (music) , algebraic number , context (archaeology) , convex set , geometry , mathematical analysis , physics , convex optimization , algorithm , paleontology , acoustics , biology
It is well known that there is no local criterion to decide the linear realizability of matroids or oriented matroids. We use the set‐up of chirotopes or oriented matroids to derive a similar result in the context of convex polytopes. There is no local criterion to decide whether a combinatorial sphere is polytopal. The proof is based on a construction technique for rigid chirotopes. These correspond, in the realizable case, to convex polytopes whose internal combinatorial structure is completely determined by its face lattice. So, a rigid chirotope is realizable over a field F if and only if its face‐lattice is F ‐polytopal. Furthermore we prove that for every proper subfield F of the field A of real algebraic numbers there exists a 6‐polytope which is not realizable over F .