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POLYTOPES
, 2013
"... Grünbaum and Shephard [40] remarked that there were three developments which foreshadowed the modern theory of convex polytopes. (1) The publication of Euclid’s Elements and the five Platonic solids. In modern terms, these are the regular 3polytopes. ..."
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Grünbaum and Shephard [40] remarked that there were three developments which foreshadowed the modern theory of convex polytopes. (1) The publication of Euclid’s Elements and the five Platonic solids. In modern terms, these are the regular 3polytopes.
Neighborly cubical polytopes
 Discrete & Computational Geometry
, 2000
"... Neighborly cubical polytopes exist: for any n ≥ d ≥ 2r + 2, there is a cubical whose rskeleton is combinatorially equivalent to that of the convex dpolytope Cn d ndimensional cube. This solves a problem of Babson, Billera & Chan. Kalai conjectured that the boundary ∂Cn d of a neighborly cubic ..."
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Cited by 18 (2 self)
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cubical polytope Cn d maximizes the fvector among all cubical (d − 1)spheres with 2n vertices. While we show that this is true for polytopal spheres if n ≤ d+1, we also give a counterexample for d = 4 and n = 6. Further, the existence of neighborly cubical polytopes shows that the graph of the n
THE UNIVERSALITY THEOREM FOR NEIGHBORLY POLYTOPES
"... Abstract. In this note, we prove that every open primary basic semialgebraic set is stably equivalent to the realization space of an evendimensional neighborly polytope. This in particular provides the final step for Mnëv’s proof of the universality theorem for simplicial polytopes. 1. ..."
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Abstract. In this note, we prove that every open primary basic semialgebraic set is stably equivalent to the realization space of an evendimensional neighborly polytope. This in particular provides the final step for Mnëv’s proof of the universality theorem for simplicial polytopes. 1.
Splitting polytopes
 Münster J. Math
"... Abstract. A split of a polytope P is a (regular) subdivision with exactly two maximal cells. It turns out that each weight function on the vertices of P admits a unique decomposition as a linear combination of weight functions corresponding to the splits of P (with a split prime remainder). This gen ..."
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Cited by 8 (5 self)
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Abstract. A split of a polytope P is a (regular) subdivision with exactly two maximal cells. It turns out that each weight function on the vertices of P admits a unique decomposition as a linear combination of weight functions corresponding to the splits of P (with a split prime remainder
NEIGHBORLY CUBICAL POLYTOPES AND SPHERES
, 2005
"... Abstract. We prove that the neighborly cubical polytopes studied by Günter M. Ziegler and the first author [14] arise as a special case of the neighborly cubical spheres constructed by Babson, Billera, and Chan [4]. By relating the two constructions we obtain an explicit description of a nonpolytop ..."
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Cited by 7 (0 self)
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Abstract. We prove that the neighborly cubical polytopes studied by Günter M. Ziegler and the first author [14] arise as a special case of the neighborly cubical spheres constructed by Babson, Billera, and Chan [4]. By relating the two constructions we obtain an explicit description of a nonpolytopal
PRODSIMPLICIALNEIGHBORLY POLYTOPES
, 2009
"... We introduce PSN polytopes whose kskeleton is combinatorially equivalent to that of a product of r simplices. They simultaneously generalize both neighborly and neighborly cubical polytopes. We construct PSN polytopes by three different methods, the most versatile of which is an extension of Sanya ..."
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Cited by 3 (0 self)
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We introduce PSN polytopes whose kskeleton is combinatorially equivalent to that of a product of r simplices. They simultaneously generalize both neighborly and neighborly cubical polytopes. We construct PSN polytopes by three different methods, the most versatile of which is an extension
Unique Minimal Liftings for Simplicial Polytopes
, 2011
"... For a minimal inequality derived from a maximal latticefree simplicial polytope in R n, we investigate the region where minimal liftings are uniquely defined, and we characterize when this region covers R n. We then use this characterization to show that a minimal inequality derived from a maximal ..."
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Cited by 6 (6 self)
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For a minimal inequality derived from a maximal latticefree simplicial polytope in R n, we investigate the region where minimal liftings are uniquely defined, and we characterize when this region covers R n. We then use this characterization to show that a minimal inequality derived from a maximal
Coordination of Groups of Mobile Autonomous Agents Using Nearest Neighbor Rules
, 2002
"... In a recent Physical Review Letters paper, Vicsek et. al. propose a simple but compelling discretetime model of n autonomous agents fi.e., points or particlesg all moving in the plane with the same speed but with dierent headings. Each agent's heading is updated using a local rule based on ..."
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Cited by 1245 (60 self)
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coordination and despite the fact that each agent's set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models.
Enumeration of neighborly polytopes and oriented matroids
, 2014
"... Neighborly polytopes are those that maximize the number of faces in each dimension among all polytopes with the same number of vertices. Despite their extremal properties they form a surprisingly rich class of polytopes, which has been widely studied and is the subject of many open problems and conj ..."
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Neighborly polytopes are those that maximize the number of faces in each dimension among all polytopes with the same number of vertices. Despite their extremal properties they form a surprisingly rich class of polytopes, which has been widely studied and is the subject of many open problems
Results 1  10
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236,972