Martin Henk, Jurgen Richter-Gebert, and Gunter M. Ziegler. Basic properties of convex polytopes. In Jacob E. Goodman and Joseph O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 13, pages 243--270. CRC Press, Boca Raton, 1997. Home/Search Document Details and Download
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Ununfoldable Polyhedra - Bern, Demaine, Eppstein, Kuo (1999) (1 citation) (Correct)
....The first example of Figure 1 is ruled out because faces of a convex polyhedron are always homeomorphic to disks. The second example is ruled out because there are pairs of faces that share more than one edge, which is impossible for a convex polyhedron 1 . In general, Steinitz s theorem [7, 10, 18] tells us that a polyhedron is topologically convex precisely if its graph is 3 connected and planar. The class of topologically convex polyhedra includes all convex faced polyhedra (i.e. polyhedra whose faces are all convex) that are homeomorphic to spheres. This will be proved formally ....
....is, polyhedra whose graphs are the graphs of convex polyhedra. A convex polyhedron is a closed polyhedron whose interior is a convex set equivalently, the open line segment connecting any pair of points on the polyhedron s surface is interior to the polyhedron. Theorem 1 (Steinitz s Theorem [7, 10, 18]) A graph is the graph of a convex polyhedron precisely if it is 3 connected and planar. Corollary 1 Every convex faced closed polyhedron that is homeomorphic to a sphere is topologically convex. Proof: Because the polyhedron is homeomorphic to a sphere, its graph Gmust be planar. It remains to ....
Martin Henk, Jurgen Richter-Gebert, and Gunter M. Ziegler. Basic properties of convex polytopes. In Jacob E. Goodman and Joseph O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 13, pages 243--270. CRC Press, Boca Raton, 1997.
Space-Time Tradeoffs for Emptiness Queries - Erickson (1997) (2 citations) (Correct)
....be formalized algebraically in the language of polyhedral cones and linear subspaces one dimension higher; we will give a less formal, purely geometric treatment. For more technical details, we refer the reader to the rst two chapters of Ziegler s lecture notes [58] or the survey by Henk et al.[36]. The # dimensional real projective space ## # can be de ned as the set of lines through the origin in the (# 1) dimensional real vector space # ### . Every # dimensional linear subspace of # ### induces a (# # 1) dimensional at # in ## # , and its orthogonal complement induces the ....
M. Henk, J. Richter-Gebert, and G. M. Ziegler, Basic properties of convex polytopes, in Handbook of Discrete and Computational Geometry, J. E. Goodman and J. O'Rourke, eds., CRC Press LLC, 1997, ch. 13, pp. 243-270.
Space-Time Tradeoffs for Emptiness Queries - Erickson (1997) (2 citations) (Correct)
....be formalized algebraically in the language of polyhedral cones and linear subspaces one dimension higher; we will give a less formal, purely geometric treatment. For more technical details, we refer the reader to the rst two chapters of Ziegler s lecture notes [58] or the survey by Henk et al.[36]. The d dimensional real projective space RP d can be de ned as the set of lines through the origin in the (d 1) dimensional real vector space R d 1 . Every k dimensional linear subspace of R d 1 induces a (k 1) dimensional at f in RP d , and its orthogonal complement induces the (d k ....
M. Henk, J. Richter-Gebert, and G. M. Ziegler, Basic properties of convex polytopes, in Handbook of Discrete and Computational Geometry, J. E. Goodman and J. O'Rourke, eds., CRC Press LLC, 1997, ch. 13, pp. 243-270.
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