Geodesic
Lawrence Polytopes
Canadian journal of mathematics, Tome 42 (1990) no. 1, pp. 62-79

Voir la notice de l'article provenant de la source Cambridge University Press

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 2n 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to prove that the class of polytopal lattices is strictly contained in the class of face lattices of oriented matroids (unpublished) and by Billera and Munson to show that the latter class is not closed under polarity.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. 
Bayer, Margaret; Sturmfels, Bernd. Lawrence Polytopes. Canadian journal of mathematics, Tome 42 (1990) no. 1, pp. 62-79. doi: 10.4153/CJM-1990-004-4
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[1] 1. Alon, N., The number of polytopes, configurations and real matroids, Mathematika 33 (1986), 62–71. Google Scholar

[2] 2. Bayer, M. and Billera, L.J., Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets, Inventiones math. 19 (1985), 143–157. Google Scholar

[3] 3. Bayer, M., The extended f-vectors of 4-polytopes, J. Combin. Theory Ser. A 44 (1987), 141–151. Google Scholar

[4] 4. Billera, L.J. and Munson, B.S., Polarity and inner products in oriented matroids, Europ. J. Combinatorics 5 (1984), 293–308. Google Scholar

[5] 5. Bland, R.G. and Vergnas, M. Las, Orientability of matroids, J. Combin. Theory Ser. B 24 (1976), 94–123. Google Scholar

[6] 6. Bokowski, J. and Sturmfels, B., Polytopal and nonpolytopal spheres: An algorithmic approach, Israel J. Math. 57 (1987), 254–271. Google Scholar

[7] 7. Bokowski, J. and Sturmfels, B., Computational synthetic geometry, Lecture Notes in Mathematics 1355 (Springer, Heidelberg, 1989). Google Scholar

[8] 8. Folkman, J. and Lawrence, J., Oriented matroids, J. Combin. Theory Ser. B 25 (1978), 199–236. Google Scholar

[9] 9. Goodman, J.E. and Pollack, R., Upper bounds for configurations and polytopes in Rd , Discrete Comput. Geometry 1 (1986), 219–227. Google Scholar

[10] 10. Grünbaum, B., Convex Polytopes (Wiley Interscience, London, 1967). Google Scholar

[11] 11. Jaggi, B., Mani-Levitska, P., Sturmfels, B. and White, N., Uniform oriented matroids without the isotopy property, Discrete Comput. Geometry 4 (1989), 97–100. Google Scholar

[12] 12. Klee, V. and Kleinschmidt, P., Polytopal complexes and their relatives, in: Handbook of combinatorics, in preparation. Google Scholar

[13] 13. Vergnas, M.Las, Convexity in oriented matroids, J. Combin. Theory Ser. B 29 (1980), 231–243. Google Scholar

[14] 14. Mandel, A., Topology of oriented matroids, Ph.D. Dissertation, University of Waterloo (1981). Google Scholar

[15] 15. McMullen, P., Transforms, diagrams and representations, in: Contributions to geometry (Birkhäuser, Basel, 1978). Google Scholar

[16] 16. Mnëv, N.E., The universality theorems on the classification problem of configuration varieties and convex poly topes varieties, in: Topology and geometry - Rohlin seminar, Lecture Notes in Mathematics 1346 (Springer, Heidelberg, 1988), 527–544. Google Scholar

[17] 17. Stanley, R.P., Enumerative combinatorics, Volume I (Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, California, 1986). Google Scholar

[18] 18. Steinitz, E. and Rademacher, H., Vorlesungen über die Théorie der Polyeder (Springer, Berlin, 1934; Reprint by Springer, Berlin, 1976) Google Scholar

[19] 19. Sturmfels, B., On the decidability of diophantine problems in combinatorial geometry, Bull. Amer. Math. Soc. 17 (1987), 121–124. Google Scholar

[20] 20. Sturmfels, B., Some applications of affine Gale diagrams to polytopes with few vertices, SIAM J. Discrete Math. 1 (1988), 121–133. Google Scholar

[21] 21. Sturmfels, B., Oriented matroids and combinatorial convex geometry, Dissertation, Technische Hochschule Darmstadt (1987). Google Scholar

[22] 22. White, N., A non-uniform matroid which violates the isotopy conjecture, Discrete Comput. Geometry 4 (1989), 1–2. Google Scholar

[23] 23. White, N., Theory of matroids, encyclopedia of math. 26 (Cambridge University Press, 1986). Google Scholar

[24] 24. Zaslavsky, T., Facing up to arrangements: face count formulas for partitions of space by hyperplanes, Memoirs Amer. Math. Soc. 154 (1975). Google Scholar

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