Simplicial neighbourly 5-polytopes with nine vertices

Finbow, Wendy

doi:10.1007/s40590-014-0013-y

Cited by 4 publications

(4 citation statements)

References 12 publications

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“…The number of combinatorial types of simplicial neighborly 5-polytopes with 9 vertices was determined by Finbow [Fin14] and also in [FMM13].…”

Section: Simplicial Neighborly 5-polytopesmentioning

confidence: 99%

Realizability and inscribability for simplicial polytopes via nonlinear optimization

Firsching

2017

Math. Program.

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We show that nonlinear optimization techniques can successfully be applied to realize and to inscribe matroid polytopes and simplicial spheres. Thus we obtain a complete classification of neighborly polytopes of dimension 4, 6 and 7 with 11 vertices, of neighborly 5-polytopes with 10 vertices, as well as a complete classification of simplicial 3-spheres with 10 vertices into polytopal and non-polytopal spheres. Surprisingly many of the realizable polytopes are also inscribable. * Supported by the DFG within SFB/TRR 109 "Discretization in Geometry and Dynamics"

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“…The number of combinatorial types of simplicial neighborly 5-polytopes with 9 vertices was determined by Finbow [Fin14] and also in [FMM13].…”

Section: Simplicial Neighborly 5-polytopesmentioning

confidence: 99%

Realizability and inscribability for simplicial polytopes via nonlinear optimization

Firsching

2017

Math. Program.

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“…This could be useful when working with finite precision, or in the problem of sparse-integer recovery ( [17]), where an integral sensing matrix is required. [16], [18] and d = 6, n = 10 [8]. In high dimension, it is known that for large enough n and d = δn with δ ∈ (0, 1) almost all randomly sampled polytopes are k-neighborly with k ≈ ρ N (d/n)d (see [14] for the definition of the neighborliness constant ρ N and more details).…”

Section: Introductionmentioning

confidence: 99%

On the neighborliness of dual flow polytopes of quivers

Gallardo,

Mckenzie

2018

Preprint

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In this note we investigate under which conditions the dual of the flow polytope (henceforth referred to as the 'dual flow polytope') of a quiver is k-neighborly, for generic weights near the canonical weight. We provide a lower bound on k, depending only on the edge connectivity of the underlying graph, for such weights. In the case where the canonical weight is in fact generic, we explicitly determine a vertex presentation of the dual flow polytope. Finally, we specialize our results to the case of complete, bipartite quivers where we show that the canonical weight is indeed generic, and are able to provide an improved bound on k. Hence, we are able to produce many new examples of high-dimensional, k-neighborly polytopes.

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“…This intrinsic difficulty has motivated the focus on the enumeration of smaller and specially interesting families of polytopes, in particular simplicial [6,31,32] and neighborly polytopes [5,7,8,15,18,19,22,31,42,57]. A d-polytope is k-neighborly if every subset of k vertices forms a face, and it is called just neighborly if it is d 2 -neighborly.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, the enumeration of neighborly polytopes is known for 4 and 6-dimensional polytopes with up to 10 vertices, as a result of the combined effort of several researchers during the 70's and 80's [5,8,15,18,19,31]. Recently, the enumeration of simplicial neighborly 5-polytopes with 9 vertices has been also completed [22,26].…”

Section: Introductionmentioning

confidence: 99%

Enumerating Neighborly Polytopes and Oriented Matroids

Miyata

Padrol

2015

Experimental Mathematics

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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 and conjectures.In this paper, we study the enumeration of neighborly polytopes beyond the cases that have been computed so far. To this end, we enumerate neighborly oriented matroids -a combinatorial abstraction of neighborly polytopes -of small rank and corank. In particular, if we denote by OM(n, r) the set of all oriented matroids of rank r and n elements, we determine all uniform neighborly oriented matroids in OM(5, ≤ 12), OM(6, ≤ 9), OM(7, ≤ 11) and OM(9, ≤ 12) and all possible face lattices of neighborly oriented matroids in OM(6, 10) and OM (8,11). Moreover, we classify all possible face lattices of uniform 2-neighborly oriented matroids in OM(7, 10) and OM (8,11). Based on the enumeration, we construct many interesting examples and test open conjectures. arXiv:1408.0688v2 [math.CO]

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Simplicial neighbourly 5-polytopes with nine vertices

Cited by 4 publications

References 12 publications

Realizability and inscribability for simplicial polytopes via nonlinear optimization

Realizability and inscribability for simplicial polytopes via nonlinear optimization

On the neighborliness of dual flow polytopes of quivers

Enumerating Neighborly Polytopes and Oriented Matroids

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