Lawrence Polytopes

Source abstract

In 1980 Jim Lawrence suggested a construction Λ which assigns to a given rank r oriented matroid M on n points a rank n + r oriented matroid Λ( M ) on 2 n points such that the face lattice of Λ( M ) is polytopal if and only if M is realizable. The Λ-construction generalized a technique used by Perles to construct a nonrational polytope [10]. It was used by Lawrence to prove that the class of polytopal lattices is strictly contained in the class of face lattices of oriented matroids (unpublished) and by Billera and Munson to show that the latter class is not closed under polarity. See [4] for a discussion of this construction and both of these applications.