Enumerating Neighborly Polytopes and Oriented Matroids

Miyata, Hiroyuki; Padrol, Arnau

doi:10.1080/10586458.2015.1015084

Cited by 9 publications

(10 citation statements)

References 53 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This implies that, for d * ≥ 3, the transversal ratio τ /n of C d * (n) is bounded by 1/2 from above. In addition to cyclic polytopes, it can be checked from the result in [15,Section 4.8] by Miyata and Padrol that the same bound works for all the neighborly oriented matroids constructed in that paper. Furthermore, all examples except for odd cycles in Section 3 satisfy 2-colorability which is an even stronger property.…”

Section: Introductionmentioning

confidence: 76%

Transversals and colorings of simplicial spheres

Briggs¹,

Dobbins²,

Lee³

2021

Preprint

View full text Add to dashboard Cite

Motivated from the surrounding property of a point set in R d introduced by Holmsen, Pach and Tverberg, we consider the transversal number and chromatic number of a simplicial sphere. As an attempt to give a lower bound for the maximum transversal ratio of simplicial d-spheres, we provide two infinite constructions. The first construction gives infintely many (d + 1)-dimensional simplicial polytopes with the transversal ratio exactly 2 d+2 for every d ≥ 2. In the case of d = 2, this meets the previously well-known upper bound 1/2 tightly. The second gives infinitely many simplicial 3-spheres with the transversal ratio greater than 1/2. This was unexpected from what was previously known about the surrounding property. Moreover, we show that, for d ≥ 3, the facet hypergraph F(P) of a (d + 1)dimensional simplicial polytope P has the chromatic number χ(F(P)) ∈ O(n), where n is the number of vertices of P. This slightly improves the upper bound previously obtained by Heise, Panagiotou, Pikhurko, and Taraz.

show abstract

Section: Introductionmentioning

confidence: 76%

Transversals and colorings of simplicial spheres

Briggs¹,

Dobbins²,

Lee³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Very recently Fukuda, Miyata and Moriyama [FMM13] classified various families of oriented matroids and obtained classification of 5-polytopes with 9 vertices. Miyata and Padrol [MP15] classified neighborly 8-polytopes with 12 vertices. Table 1 summarizes known and new enumeration results of families of d-polytopes on n vertices.…”

Section: Previous Resultsmentioning

confidence: 99%

“…We combine these techniques with previous classification results: For the enumeration of some families of neighborly polytopes we build on the enumeration of corresponding families of neighborly uniform oriented matroids given by Miyata and Padrol [MP15] [Miy] For the enumeration of simplicial 4-polytopes with 10 vertices and 4-polytopes with small valence we build on the enumeration of corresponding simplicial spheres by Lutz [Lut08] [FLS] [Lut].…”

Section: Our Contributionsmentioning

confidence: 99%

“…Our notation mostly coincides with those outlined in [BLVS + 99], [RGZ04, Sect. 6], [BS89] and the introduction of [MP15].…”

Section: Definitions and Notationsmentioning

confidence: 99%

“…Miyata and Padrol [MP15] enumerate simplicial neighborly uniform oriented matroids of various ranks and number of elements. This allows us to apply the methods from Section 2 in order to find realizations of those neighborly uniform oriented matroids.…”

Section: Realizations and Inscriptions Of Neighborly Polytopesmentioning

confidence: 99%

See 2 more Smart Citations

Realizability and inscribability for simplicial polytopes via nonlinear optimization

Firsching

2017

Math. Program.

View full text Add to dashboard Cite

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"

show abstract

Six Topics on Inscribable Polytopes

Padrol

Ziegler

2016

Advances in Discrete Differential Geometry

Self Cite

View full text Add to dashboard Cite

We discuss six topics related to inscribable polytopes, both in dimension 3 (where the topic was started with a problem posed by Steiner in 1832) and in higher dimensions. It asks whether every (3-dimensional) polytope is inscribable; that is, whether for every 3-polytope there is a combinatorially equivalent polytope with all the vertices on the sphere. And if not, which are the cases of 3-polytopes that do have such a realization? He also asks the same question for circumscribable polytopes, those that have a realization with all the facets tangent to the sphere, as well as for other surfaces of degree 2.There was no progress on this question until 1928, when Ernst Steinitz showed that inscribability and circumscribability are polar concepts and presented the first A.

show abstract

Enumerating Neighborly Polytopes and Oriented Matroids

Cited by 9 publications

References 53 publications

Transversals and colorings of simplicial spheres

Transversals and colorings of simplicial spheres

Realizability and inscribability for simplicial polytopes via nonlinear optimization

Six Topics on Inscribable Polytopes

Contact Info

Product

Resources

About