Weakly Regular Subdivisions
Tóm tắt
It is shown that 2-dimensional subdivisions can be made regular by moving their vertices within parallel 1-dimensional spaces. As a consequence, any 2-dimensional subdivision is projected from the boundary complex of a 4-polytope.
Tài liệu tham khảo
Barnette, D.W.: Projections of 3-polytopes. Isr. J. Math. 8, 304–308 (1970)
Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. Classics in Applied Mathematics, vol. 9. SIAM, Philadelphia (1994)
Billera, L.J., Sturmfels, B.: Fiber polytopes. Ann. Math. 135, 527–549 (1992)
Billera, L.J., Kapranov, M.M., Sturmfels, B.: Cellular strings on polytopes. Proc. Am. Math. Soc. 122, 549–555 (1994)
Edelsbrunner, H.: An acyclicity theorem for cell complexes in d dimension. Combinatorica 10, 251–260 (1990)
Ferrez, J.-A., Liebling, Th.M.: Dynamic triangulations for efficient detection of collisions between spheres with applications in granular media simulations. Philos. Mag., B 82, 905–929 (2002)
Lawson, C.L.: Software for C 1 interpolation. In: Rice, J. (ed.) Mathematical Software III, pp. 161–194. Academic Press, New York (1977)
Lee, C.W.: Regular triangulations of convex polytopes. In: Gritzmann, P., Sturmfels, B. (eds.) Applied Geometry and Discrete Mathematics—The Victor Klee Festschrift. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 4, pp. 443–456. American Mathematical Society, Providence (1991)
Lee, C.W.: Subdivisions and triangulations of polytopes. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, pp. 383–406. CRC Press, Boca Raton (2004)
Maxwell, J.C.: On reciprocal figures and diagrams of forces. Philos. Mag. Ser. 7 27, 250–261 (1864)
Maxwell, J.C.: On reciprocal figures frames, and diagrams of forces. Trans. R. Soc. Edinb. 26, 1–40 (1869)
Pournin, L., Liebling, Th.M.: Constrained paths in the flip-graph of regular triangulations. Comput. Geom. 37, 134–140 (2007)
Pournin, L., Liebling, Th.M.: From spheres to spheropolyhedra: generalized Distinct Element Methodology and algorithm analysis. In: Cook, W., Lovász, L., Vygen, J. (eds.) Research Trends in Combinatorial Optimization, pp. 347–363. Springer, Berlin (2009)
Pournin, L.: A result on flip-graph connectivity. Adv. Geom. (in press)
Rudin, M.E.: An unshellable triangulation of a tetrahedron. Bull. Am. Math. Soc. 64, 90–91 (1958)
Rybnikov, K.: Stresses and liftings of cell-complexes. Discrete Comput. Geom. 21, 481–517 (1999)
Santos, F.: A point set whose space of triangulations is disconnected. J. Am. Math. Soc. 13, 611–637 (2000)
Santos, F.: On the refinements of a polyhedral subdivision. Collect. Math. 52, 231–256 (2001)
Santos, F.: Geometric bistellar flips: the setting, the context and a construction. In: Sanz-Sole, M., Soria, J., Varona, J.L., Verdera, J. (eds.) Proceedings of the ICM 2006, vol. III, pp. 931–962. Eur. Math. Soc., Zurich (2006)
Steinitz, E.: Polyeder und Raumeinteilungen. Enzykl. Math. Wiss. 3, 1–139 (1922)
Whiteley, W.: Motions and stresses of projected polyhedra. Topol. Struct. 7, 13–38 (1982)
Xue, X.J., Righetti, F., Telley, H., Liebling, Th.M., Mocellin, A.: The Laguerre model for grain growth in three dimensions. Philos. Mag., B 75, 567–585 (1997)