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### Is every planar point set be projections of vertices of a neighborly 4-polytope?

More exactly, written in coordinates, I'm curious to know if for every point set $(x_i,y_i)$ there are $(x_i,y_i,z_i,w_i)$ be vertices of a neighborly polytope.
This problem comes from a simple ...

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### The Universality Theorem by Mnev for uniform oriented matroids of rank 4 and higher

According to the Universality Theorem by Mnev (see below theorem 8.6.6 from [1]), for any open semialgebraic variety V there is a uniform oriented matroid of rank 3 whose realization space is stably ...

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### What are / could be the applications of Delaunay oriented matroids?

The Delaunay oriented matroid is studied in detail by F. Santos (actually this paper is more general).
For a set of points $S$ (in any dimension), let $C$ be a sphere, $C^+$ be its interior, $C^-$ be ...

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### Small remarkable matroids

Working on phased matroids (a generalization of oriented matroid to the complex case) I've found an interesting formula for computing the inner Tutte group (and, hence, all the Tutte groups) ...

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### The lattice of covectors of an oriented matroid

Let $M$ be an oriented matroid on the ground set $E$, and let $L(M)$ be its ranked poset of covectors. By definition, $L(M)$ is a sub-poset of the poset $\{0, \pm 1\}^E$, ordered by putting $0 < ...

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### Positroids and Totally Nonnegative Complex Grassmanian

Recently I begin working on matroids, in particular to a generalization of oriented matroids to the complex case.
I found on arxiv the following interesting articles:
1)Alexander Postnikov: Total ...

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### Circuits in a linear oriented matroid

Given $E$ a finite subset of a real vector space $V$, a circuit of the associated matroid is a minimal linearly dependent subset of $E$. For each circuit $\underline C$, a minimal linear dependence
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### Functionals on oriented matroids

Oriented matroids are abstractions of hyperplane arrangements, or equivalently vector configurations. Let me recall the definition in terms of covectors.
Let $R=\lbrace 0,+,-\rbrace$ with the monoid ...

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### Realizability of extensions of a free oriented matroid by an independent set

Question:
I am searching for a non-realizable matroid with few dependencies relative to the number of points. Precisely, I would like to find a non-realizable (over $\mathbb{R}$) oriented matroid $M$ ...

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### Triangulations of polytopes and tilings of zonotopes

Consider a set $A = \{ a_1,a_2,\ldots, a_n \} $ of vectors in $\mathbb{R}^d$, which lie in a common affine hyperplane. Two convex polytopes may be obtained from $A$, namely the convex hull of the ...