# Questions tagged [hyperplane-arrangements]

A hyperplane arrangement is a set of hyperplanes in a vector space or in a projective space. The complement of the union of these hyperplanes defines an algebraic variety, with interesting geometry and topology.

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### Number of regions formed by $n$ points in general position

Given $n$ points in $\mathbb{R}^d$ in general position, where $n\geq d+1$. For every $d$ points, form the hyperplane defined by these $d$ points. These hyperplanes cut $\mathbb{R}^d$ into several ...

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### Regions of hyperplane arrangements and their faces

Consider a finite hyperplane arrangement $\mathcal{A}$ over $\mathbb{R}^n$. Let the regions given by $\mathcal{A}$ be $\mathcal{R}(\mathcal{A})=\{A_1,\dots A_m\}$ for some $m$.
For any index set $I\...

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### Separation in $l^1$ (Kreps-YanTheorem)

I have a question about the hypotheses of the Kreps-Yan Separation Theorem. I use the notation $l^p_+$ for the subspace of vectors all of whose coordinates are non-negative and define $l^p_- = -l^p_+$....

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### Simplicial set represented by an (unordered) set

Let $X$ be a (finite if you want) set and form the simplicial set $F^{\bullet}(X)$ with
$$
F^{n}(X) = \mathrm{Hom}_{\mathrm{set}} ([n], X)
$$
where the right hand side denotes arbitrary maps of sets (...

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### Homotopy type of hyperplane arrangements intersected with real subspaces

The homotopy type, and especially the higher homotopy groups of complement of hyperplane arrangements in $\mathbb{C}^n$ has been extensively studied, for example Falk and Randell - On the homotopy ...

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### What are the “ouverts convenables” used to prove Brieskorns lemma?

In the proof of Brieskorns lemma, see 3.3 here, Brieskorn mentions that we take "ouverts convenables" satisfying some properties, but, as far as I can tell, never specifies what these opens actually ...

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### Seeking a more symmetric realization of a configuration of 10 planes, 25 lines and 15 points in projective space

I've got ten (projective) planes in projective 3-space:
\begin{align}
&x=0\\
&z=0\\
&t=0\\
&x+y=0\\
&x-y=0\\
&z+t=0\\
&x-y-z=0\\
&x+y+z=0\\
&x-y+t=0\\
&x+y-t=0
...

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### Random walks in arrangements of lines in the plane

Let $\cal{A}_n$ be a simple arrangement of $n$ lines in $\mathbb{R}^2$.
(Simple: each pair of lines meet in a distinct point, i.e.,
no three lines pass through the same point.)
Start a random walk at ...

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### Combinatorial region-halfplane incidence structures

I've seen a bunch of similar MO questions, yet hopefully this is not a complete duplicate.
Consider $n$ halfplanes in $\mathbb{R}^2$ with their borders in general position, that is, no point of $\...

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### The characteristic varieties of the complement of the braid arrangement

The characteristic varieties $V_d^i(X)$ of a (sufficiently nice) space $X$ are the cohomology jumping loci for 1-dimensional (complex) local systems on $X$. Assume that $H_1(X;\mathbb{Z}) \cong \...

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### Counting hyperplane arrangements up to combinatorial equivalence, simple examples and history

Two arrangements of (affine) hyperplanes in $d$-dimensional Euclidean space are combinatorially isomorphic (or combinatorially equivalent) if they have isomorphic posets of faces.
Counting the ...

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### What breaks down in the theory of affine hyperplane arrangments?

It appears to me that there is a substantial amount of combinatorial algebra and geometry supporting the theory of central hyperplane arrangements (See Topics in Hyperplane Arrangements, Aguiar and ...

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### Interactions between pseudoline arrangements and braid groups?

It is common to represent
pseudoline arrangements
as wiring diagrams:
Fig. from: "Hamiltonicity and colorings of arrangement ...

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### Singularities at worst like a hyperplane arrangement

Is there a standard name for the type of singularities a codimension-$1$ subvariety of a smooth algebraic variety has when it looks locally (possibly analytically) like an arrangement of hyperplanes? «...

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### Why does the type-A subdivision algebra look like the Rota-Baxter algebra axiom?

Let $\mathbf{k}$ be a commutative ring, and $\beta$ an element of $\mathbf{k}$. Fix a positive integer $n$, and set $\left[n\right] = \left\{1,2,\ldots,n\right\}$.
The $n$-th type-A subdivision ...

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### Are the following hyperplane arrangements previously studied?

For a subset $I$ of $[n]$, a hyperplane $H_I \subset \mathbb{R}^n$ is defined by $$\sum_{i \in I} x_i= \sum_{j \not\in I} x_j.$$
Have you seen the following hyperplane arrangements? Is there anything ...

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### Bounds on k-tuple points for intersections of hyperplanes

Suppose that $H_1$,...,$H_d$ are hyperplanes in $\mathbb P^n$ (over some field -- you can pick). For $k \geq n$, let $t_k$ denote the number of points through which there pass exactly $k$ hyperplanes....

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### Rigid line arrangements

What is already known about rigid line arrangements? By line arrangement, I mean a unions of lines in $\mathbb{P}^2_{\mathbb{C}}$ with fixed incidences. (Written in notation, I mean a collection of ...

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### Polyhedral structure of functions writable as a finite signed sum of max of linear functions

For any two positive integers $k,n$ consider the space of functions writable as,
$\sum_i \sigma_i \max \{ L_{i1},L_{i2},..,L_{ik} \}$ (a finite sum) where each $L_{*} : \mathbb{R}^n \rightarrow \...

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### Bijection directly from (n,n+1)-core partitions to parking functions?

It is well-known that the increasing parking functions are counted by the Catalan numbers. The Catalan numbers also count the dominant alcoves in the Shi arrangement of type $A_{n}$. Athanasiadis-...

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### Counting the polytopes of the translates of the resonance hyperplane arrangement inside the unit hypercube

Let $n$ be a positive natural number. For all $\emptyset \subset S \subseteq \{1, \ldots, n\}$ and $k \in \mathbb{Z}$, define the hyperplane $H(S,k)$ in $\mathbb{R}^n$ given by the equations
$$H(S,k):=...

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### Is there invariant for regions of central hyperplane arrangement?

Consider central hyperplane arrangement A with normal vectors with all combinations of -1 and 1 (is there name for it?).
There is simple invariant for each chamber of A: sum of vectors, corresponding ...

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### What are Sylvester-Gallai configurations in the complex projective plane?

A Sylvester-Gallai configuration in the the complex projective plane is a finite number of $n\ge 2$ points in the complex projective plane such that there is no line through exactly two of them. ...

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### Chromatic number of a graph defined by $n$ lines on the plane

Given $n$ lines on the plane, consider all their intersection points. Find the minimal number $d=d(n)$ such that they may be always colored in $d$ colors so that on each line any two consecutive ...

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### Affine Hyperplane Arrangements in $\mathbb R^d$

Consider $\mathcal A=(u_i)_{i=1}^m $ to be a set of hyperplanes in $\mathbb R^d$, such that for every $1\leq i \leq m$: $u_i \in \mathbb R^d$.
These hyperplanes are disconnecting $\mathbb R^d$ to ...

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319 views

### Counting Regions in Hyperplane Arranglements

Consider the following:
1) How many connected regions can $n$ hyperplanes form in $\mathbb R^d$?
2) What if the set of hyperplanes are homogeneous?
3) Given a set of $n$ pairs of ...

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### Is there any notion of a slack-matrix (and its rank) for a hyperplane arrangement?

I am assuming that the common lineality space of the hyperplane arrangement has already been factored out. So we are looking at an ``effective" arrangement in lower dimensions where all the polyhedra ...

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### Chambers of central hyperplane arrangements

Let $\mathcal{A}$ be a central hyperplane arrangement in a (finite dimensional) real vector space $V$. Assume for each hyperplane $H\in\mathcal{A}$ that we're given a labelling $H^+$, $H^-$ of the ...

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### Counting problems where unlabeled is easier than labeled

I was encouraged to post this question by Jim Propp during a meeting of the Cambridge Combinatorics and Coffee Club. It is a counterpoint to the MathOverflow question "Counting Problems where Labeled ...

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### Characterizing normal vectors of affinely independent subsets of the hypercube

Suppose we are looking at the hypercube in $\mathbb{R}^n$. I tend to the think of the cube with vertex coordinates 0 or 1, but maybe this is easier for the $\pm1$ cube.
Now suppose we have an affine ...

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653 views

### Maximal Number of Pairs of Orthogonal vectors in a set of $n$ vectors in $\mathbb{R}^3$

Suppose you are given a set of $n$ non-zero vectors in $\mathbb{R}^3$. What is the maximum number of pairs of them that are orthogonal? The current guess is $\le 2n$.
EDIT: I forgot to add that no ...

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### Open problems in hyperplane/subspace arrangements?

What are some open problems in hyperplane/subspace arrangements, preferably of the combinatorial algebraic topology kind, and where can one read about them? That is, where are they discussed, and ...

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### Volume of bounded regions in hyperplane arrangements

I am given a hyperplane arrangement $\mathcal{H}_0$ in $\mathbb{R}^n$ and a function $\phi \colon \mathbb{R}^n \to \mathbb{Q}.$ I choose any enumeration on the set of primitive vectors (i.e. vectors ...

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### Number of regions of a hyperplane arrangement avoiding a generic hyperplane

Let $\mathcal{A}$ be an essential arrangement of hyperplanes in $\mathbb{R}^n$. Zaslavsky's theorem says that the number of regions of $\mathcal{A}$ is given by $r(\mathcal{A})=(-1)^n\chi_{\mathcal{A}}...

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### Characterizing orthants with polynomials

Let $x=(x_1,\ldots,x_n)\in\mathbb{R}^n$. Can one find a polynomial $p$ (of arbitrary degree) in the coordinates of $x$ such that $p(x)\geq 0$ if and only if $x$ is an element of the positive orthant $\...

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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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### help with cohomology of $\mathbb{P}^n$ relative to a NCD

Let $H_0, \ldots, H_n$ be $n$ hyperplanes in $\mathbb{P}^n(\mathbb{C})$ with normal crossings and denote by $H$ the union of them. I am trying to understand why
(1) $H^n(\mathbb{P}^n(\mathbb{C}), H)$...

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### Tensor product of hyperplane arrangements

Let $(A_{1},V_{1})$ and $(A_{2},V_{2})$ be two central hyperplane arrangements, which 0 belongs to their intersections. Let $V=V_{1}\otimes V_{2}$. Define the tensor product arrangement $(A_{1}\otimes ...

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### Extending a line-arrangement so that the bounded components of its complement are triangles

Given a finite collection of lines $L_1,\dots,L_m$ in ${\bf{R}}^2$, let $R_1,\dots,R_n$ be the connected components of ${\bf{R}}^2 \setminus (L_1 \cup \dots \cup L_m)$, and say that {$L_1,\dots,L_m$} ...

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### How many regions are created by the set of all hyperplanes defined by a set of points?

If we have a set of points X in d-dimensional euclidean space, and we look at the set of all hyperplanes that are defined by any subset Y of X (in the sense of being the unique hyperplane containing ...

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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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### Is there a characterization of hyperplane arrangement intersection posets?

For a hyperplane arrangement $\mathcal{A}$ over a vector space $V$, we define its intersection poset, $L(\mathcal{A})$, as the set of all nonempty intersections of hyperplanes in $\mathcal{A}$ ordered ...

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### lines through A_n reflection arrangement and permutations

(updated; apologies for way too much room left for interpretation in the original post)
Let $\mathcal{A} =A_{n-1}$ be the $A_{n-1}$ arrangement in $\mathbb{R}^{n}$, i.e. the set of hyperplanes $H_{ij}...

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### Hyperplane arrangements and covering numbers

Let $H$ be a set of $(d-1)$-dimensional hyperplanes in $\mathbb{R}^d$. For each hyperplane $h \in H$ let $D(h)$ and $\bar{D}(h)$ be the corresponding half spaces of $\mathbb{R}^d$. For a point $x \...

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511 views

### Maximal disjoint hyperplanes

Assume a set of $n^{r}$ points $X_{r} = \{ x_{1}, \cdots, x_{n^{r}} \}$ is given occupying a codimension $t^{r}$ subspace in $\mathbb{R}^{n^{r}}$. Let $M_{r}$ be the set of $t^{r}$-tuples of these ...

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### A natural refinement of the $A_n$ arrangement is to consider all $2^n-1$ hyperplanes given by the sums of the coordinate functions. Have you seen this arrangement? Is it completely intractable?

The short version
Here is an extremely natural hyperplane arrangement in $\mathbb{R}^n$, which I will call $R_n$ for resonance arrangement.
Let $x_i$ be the standard coordinates on $\mathbb{R}^n$. ...

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### balls in arrangements of hyperplanes

The following theorem is from Aronov, Naiman, Pach and Sharir's
An invariant property of balls in arrangements of hyperplanes. I would like to state them and then ask if any related problem/theorem ...

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### How does one map regression depth to undirected depth of a point?

The regression depth of a line is the minimum number of points it has to cross to take it from its initial position to vertical. The undirected depth of a point is the minimum number of lines a ray ...

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### Criterion for being a simple vector

1) I was wondering if there exists a criterion (of (a) combinatorial or (b) geometric nature) for a sum of simple vectors $V\in\wedge^k(\mathbb R^n)$,
$V=e_{a_{11}}\wedge\cdots\wedge e_{a_{1k}} + \...

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796 views

### Coxeter Arrangements and an Identity

Let $\{A_i\}$ be a collection of $m$ hyperplanes in $\mathbb{C}^n$ which all pass through the origin (a central hyperplane arrangement). Such an arrangement is called Coxeter if reflecting across any ...