G. Kalai, Some aspects in the combinatorial theory of convex polytopes, in [18], pp. 205-230. Home/Search Document Details and Download
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Inner Diagonals Of Convex Polytopes - Bremner, Klee (1998) (1 citation) (Correct)
....1) simplex) is v 2 dv d 1 2 , and this bound is attained by precisely the triangulated manifolds that are isomorphic to the boundary complex of a stacked polytope. In order to generalize statement (L2) to not necessarily simplicial polytopes, we need the following theorem of Kalai [Ka2]. Let P F denote the quotient polytope of P with respect to F , i.e. a polytope whose face lattice is isomorphic to the interval G F # G # P of the face lattice of P (see [Zi] p. 57) 3.7. Theorem (Kalai) For any d polytope P and any face F of P , g 2 (P ) # g 2 (F ) g 1 (F )g ....
G. KALAI, Some aspects of the combinatorial theory of convex polytopes, Polytopes: Abstract, Convex and Computational., T. Bisztriczky, P. McMullen, R. Schneider, and A. Ivic Weiss, editors, Kluwer, Dordrecht, 1993, pp. 205--229.
Inner diagonals of convex polytopes (Extended Abstract) - Bremner, Klee (Correct)
.... (not necessarily simplicial) polytopes by Stanley [11] Kalai [8] generalized this to all polytopes by using notions of rigidity of graphs, in particular a theorem of Whitely [14] In order to generalize statement (L2) to not necessarily simplicial polytopes, we need the following theorem of Kalai [9]. Let P F denote the quotient polytope of P with respect to F , i.e. a polytope whose face lattice is isomorphic to the interval G F # G # P of the face lattice of P (see Ziegler [15] p. 57) 3 4.4. Theorem (Kalai) For any d polytope P and any face F of P , g2(P ) # g2(F ) ....
G. Kalai. Some aspects of the combinatorial theory of convex polytopes, pages 205--229. Kluwer, Dordrecht, 1993.
Inner Diagonals Of Convex Polytopes - Bremner, Klee (1998) (1 citation) (Correct)
.... Delta Gamma dv Gamma d 1 2 Delta , and this bound is attained by precisely the triangulated manifolds that are isomorphic to the boundary complex of a stacked polytope. In order to generalize statement (L2) to not necessarily simplicial polytopes, we need the following theorem of Kalai [Ka2]. Let P=F denote the quotient polytope of P with respect to F , i.e. a polytope whose face lattice is isomorphic to the interval f G j F G Pg of the face lattice of P (see [Zi] p. 57) 3.7. Theorem (Kalai) For any d polytope P and any face F of P , g 2 (P ) g 2 (F ) g 1 (F )g 1 (P=F ) ....
G. KALAI, Some aspects of the combinatorial theory of convex polytopes, Polytopes: Abstract, Convex and Computational., T. Bisztriczky, P. McMullen, R. Schneider, and A. Ivi'c Weiss, editors, Kluwer, Dordrecht, 1993, pp. 205--229.
Polytope Skeletons And Paths - Kalai (1997) (1 citation) Self-citation (Kalai) (Correct)
....number of combinatorial types of k skeletons of d polytopes with d b 1 vertices. Then, for xed b and k, f(d; k; b) is bounded. This follows from The number of empty i pyramids for d polytopes with d b vertices is bounded by a function of i and b. For another proof of this theorem see [58]. For a d polytope P , let e i (P ) denote the number of empty i simplices of P . Characterize the sequence of numbers (e 1 (P ) e 2 (P ) e d (P ) arising from simplicial d polytopes and from general d polytopes. The following theorem motivated by commutative algebraic problems ....
G. Kalai, Some aspects in the combinatorial theory of convex polytopes, in [18], pp. 205-230.
Algebraic Shifting - Kalai (2001) Self-citation (Kalai) (Correct)
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G. Kalai, Some aspects in the combinatorial theory of convex polytopes, in : Polytopes, Abstract Convex and Computational, (T. Bisztriczky et alls, eds) pp. 205-230, 1995.
Combinatorics with a Geometric Flavor: Some Examples - Kalai (2000) (1 citation) Self-citation (Kalai) (Correct)
....mentioned above. The result of Braden and MacPherson is a sharpening as well as a farreaching generalization of (4.4) for general polytopes. It is proved, however, only for rational polytopes. I will now state this result without explaining properly the background and I refer the reader to [51, 81, 63] for more. For a d polytope P let h P (x) d X k=0 h i (P )x k ; g P (x) d=2] X i=0 g k (P )x k : Here h k (P ) dim IH 2k (T P ) and g k (P ) h k (P ) h k 1 (P ) where T P is the toric variety associated to P and IH is intersection homology. The quantities dim IH k (T P ) ....
....and probably also for polytopes. Adin [35] found the right notion of h numbers, but a construction for a cubical Stanley Reisner ring is yet unknown. 4. 7 Some links and references Polytope theory [89, 58, 59, 42, 66, 46, 78] Ziegler] face numbers and h numbers of polytopes and complexes [43, 48, 45, 80, 63], open problems [85] cubical spheres and polytopes [38, 60] a continuous version of the UBT [87] Kuhnel s CP 2 and other special triangulations [70, 69, 68] Kruskal Katona theorem and related results [54] Turan type theorems [49, 55] commutative algebra and combinatorics [82, 53] Herzog] ....
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G. Kalai, Some aspects in the combinatorial theory of convex polytopes, in
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