# 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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### 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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### 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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### 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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### 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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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 $x \... 0answers 511 views ### Maximal disjoint hyperplanes Assume a set of$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}$-tuples of these ... 1answer 986 views ### A natural refinement of the$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$\mathbb{R}^n$. ... 0answers 266 views ### 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 ... 2answers 158 views ### 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 ... 1answer 251 views ### 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}} + \...
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 ...