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127
Dynamical partitions of space in any dimension
, 1998
"... Abstract. Topologically stable cellular partitions of Ddimensional spaces are studied. A complete statistical description of the average structural properties of such partitions is given in terms of a sequence of 2 ) variables for D even (or odd). These variables are the average coordination numbe ..."
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numbers of the 2kdimensional polytopes (2k < D) which make up the cellular structure. A procedure to produce Ddimensional space partitions through celldivision and cellcoalescence transformations is presented. Classes of structures which are invariant under these transformations are found
Dynamical partitions of space in any dimension
, 1998
"... Abstract. Topologically stable cellular partitions of Ddimensional spaces are studied. A complete statistical description of the average structural properties of such partitions is given in terms of a sequence of D2 − 1 (or D−12) variables for D even (or odd). These variables are the average coordi ..."
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coordination numbers of the 2kdimensional polytopes (2k < D) which make up the cellular structure. A procedure to produce Ddimensional space partitions through celldivision and cellcoalescence transformations is presented. Classes of structures which are invariant under these transformations are found
Central limit theorems for random polytopes in convex polytopes
, 2005
"... Let K be a smooth convex set. The convex hull of independent random points in K is a random polytope. Central limit theorems for the volume and the number of i dimensional faces of random polytopes are proved as the number of random points tends to infinity. One essential step is to determine the pr ..."
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Cited by 31 (8 self)
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Let K be a smooth convex set. The convex hull of independent random points in K is a random polytope. Central limit theorems for the volume and the number of i dimensional faces of random polytopes are proved as the number of random points tends to infinity. One essential step is to determine
Polytopes in Arrangements
, 1999
"... Consider an arrangement of n hyperplanes in R d . Families of convex polytopes whose boundaries are contained in the union of the hyperplanes are the subject of this paper. We aim to bound their combinatorial complexity. Exact asymptotic bounds were known for the case where the polytopes are cells ..."
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Cited by 2 (1 self)
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vertices. In R 3 we show an O(k 1=3 n 2 ) bound on the number of faces of k such polytopes. We also discuss worstcase lower bounds and higherdimensional versions of the problem. Among other results, we show that the maximum number of facets of k pairwise vertexdisjoint polytopes in R d is k 1
ON THE GRAPHCONNECTIVITY OF SKELETA OF CONVEX POLYTOPES
, 2008
"... Given a ddimensional convex polytope P and nonnegative integer k not exceeding d − 1, let Gk(P) denote the simple graph on the node set of kdimensional faces of P in which two such faces are adjacent if there exists a (k + 1)dimensional face of P which contains them both. The graph Gk(P) is isom ..."
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Cited by 3 (2 self)
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Given a ddimensional convex polytope P and nonnegative integer k not exceeding d − 1, let Gk(P) denote the simple graph on the node set of kdimensional faces of P in which two such faces are adjacent if there exists a (k + 1)dimensional face of P which contains them both. The graph Gk
Linkages in Polytope Graphs
, 2007
"... A graph is klinked if any k disjoint vertexpairs can be joined by k disjoint paths. We improve a lower bound on the linkedness of polytopes slightly, which results in exact values for the minimal linkedness of 7, 10 and 13dimensional polytopes. We analyze in detail linkedness of polytopes on at ..."
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A graph is klinked if any k disjoint vertexpairs can be joined by k disjoint paths. We improve a lower bound on the linkedness of polytopes slightly, which results in exact values for the minimal linkedness of 7, 10 and 13dimensional polytopes. We analyze in detail linkedness of polytopes
Toric Varieties and Lattice Polytopes
, 2006
"... We will show how to construct spaces called toric varieties from lattice polytopes. Toric fibrations correspond to slices of the polytope, and the Dynkin diagrams of singularities will appear in the lattice polytope. This process has been used to study K3 surfaces and higherdimensional CalabiYau m ..."
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We will show how to construct spaces called toric varieties from lattice polytopes. Toric fibrations correspond to slices of the polytope, and the Dynkin diagrams of singularities will appear in the lattice polytope. This process has been used to study K3 surfaces and higherdimensional Calabi
Flag vectors of multiplicial polytopes
 Electron. J. Combin., 11(1):Research Paper
, 2004
"... Bisztriczky introduced the multiplex as a generalization of the simplex. A polytope is multiplicial if all its faces are multiplexes. In this paper it is proved that the flag vectors of multiplicial polytopes depend only on their face vectors. A special class of multiplicial polytopes, also discover ..."
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Cited by 3 (2 self)
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− k) and characteristic k: hi = � k−d+i i i−1 �,fori ≤ d/2. In addition, a construction is given for 4dimensional multiplicial polytopes having twothirds of their vertices on a single facet, answering a question of Bisztriczky. 1
THE GROUPS OF THE REGULAR STARPOLYTOPES
, 1998
"... The regular starpolyhedron f5Ò 52g is isomorphic to the abstract polyhedron f5Ò 5 j3g, where the last entry “3” in its symbol denotes the size of a hole, given by the imposition of a certain extra relation on the group of the hyperbolic honeycomb f5Ò 5g. Here, analogous formulations are found fo ..."
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Cited by 3 (1 self)
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for the groups of the regular 4dimensional starpolytopes, and for those of the nondiscrete regular 4dimensional honeycombs. In all cases, the extra group relations to be imposed on the corresponding Coxeter groups are those arising from “deep holes”; thus the abstract description of f5Ò 3kÒ 52g is f5Ò 3kÒ 5
How neighborly can a centrally symmetric polytope be
 Discrete and Computational Geometry
"... Abstract. We show that there exist kneighborly centrally symmetric ddimensional polytopes with 2(n + d) vertices, where ..."
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Cited by 31 (4 self)
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Abstract. We show that there exist kneighborly centrally symmetric ddimensional polytopes with 2(n + d) vertices, where
Results 1  10
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127