Complete enumeration of small realizable oriented matroids
TLDR
In this paper, the authors investigated algorithmic ways to classify oriented matroids in terms of realizability, and determined all possible combinatorial types (including degenerate ones) of 3-dimensional configurations of 8 points, 2-dimensional configuration of 9 points, and 5-dimensional polytopes with nine vertices.Abstract:
Enumeration of all combinatorial types of point configurations and polytopes is a fundamental problem in combinatorial geometry. Although many studies have been done, most of them are for 2-dimensional and non-degenerate cases. Finschi and Fukuda (Discrete Comput Geom 27:117–136, 2002) published the first database of oriented matroids including degenerate (i.e., non-uniform) ones and of higher ranks. In this paper, we investigate algorithmic ways to classify them in terms of realizability, although the underlying decision problem of realizability checking is NP-hard. As an application, we determine all possible combinatorial types (including degenerate ones) of 3-dimensional configurations of 8 points, 2-dimensional configurations of 9 points, and 5-dimensional configurations of 9 points. We also determine all possible combinatorial types of 5-polytopes with nine vertices.read more
Citations
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Journal ArticleDOI
Realizability and inscribability for simplicial polytopes via nonlinear optimization
TL;DR: It is shown that nonlinear optimization techniques can successfully be applied to realize and to inscribe matroid poly topes and simplicial spheres and many of the realizable polytopes are also inscribable.
Journal ArticleDOI
The Geometry of Gaussoids
TL;DR: In this article, the Lagrangian Grassmannian and its symmetries are used to define gaussoids, a combinatorial structure that encodes independence in probability and statistics, just like matroids encode independence in linear algebra.
Journal ArticleDOI
Enumerating Neighborly Polytopes and Oriented Matroids
Hiroyuki Miyata,Arnau Padrol +1 more
TL;DR: The enumeration of neighborly polytopes of small rank and corank is studied beyond the cases that have been computed so far and many interesting examples are constructed and open conjectures are tested.
DissertationDOI
Optimization Methods in Discrete Geometry
TL;DR: A simplicial neighborly polytopes and self-dual 2-colored necklaces were used in this paper for the realization of simplicial spheres and oriented matroids.
Journal ArticleDOI
Enumeration of 2-Level Polytopes
Adam Bohn,Yuri Faenza,Samuel Fiorini,Vissarion Fisikopoulos,Marco Macchia,Kanstantsin Pashkovich +5 more
TL;DR: The approach is based on the notion of a simplicial core, that allows the problem to be reduced to the enumeration of the closed sets of a discrete closure operator, along with some convex hull computations and isomorphism tests.
References
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Book
Lectures on Polytopes
TL;DR: In this article, the authors present a rich collection of material on the modern theory of convex polytopes, with an emphasis on the methods that yield the results (Fourier-Motzkin elimination, Schlegel diagrams, shellability, Gale transforms, and oriented matroids).
BookDOI
Ideals, Varieties, and Algorithms
TL;DR: In the Groebner package, the most commonly used commands are NormalForm, for doing the division algorithm, and Basis, for computing a Groebners basis as mentioned in this paper. But these commands require a large number of variables.
Quantifier elimination for real closed fields by cylindrical algebraic decomposition
TL;DR: In this paper, a quantifier elimination method for the elementary theory of real closed fields is presented. But it does not provide a decision method, which enables one to decide whether any sentence of the theory is true or false, since many important and difficult mathematical problems can be expressed in this theory.
Journal ArticleDOI
The number of faces of a simplicial convex polytope
TL;DR: In this paper, the necessity of McMullen's condition on the f-vector of a simplicial convex d-polytope was shown to be complete and sufficient for f = (f., fi,..., fd...J of integers).