**12**

votes

**1**answer

434 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 ...

**3**

votes

**0**answers

61 views

### Details on the Symmetric Group action on chambers of the Shi Arrangement

In "A Simple Bijection for the Regions of the Shi Arrangement of Hyperplanes", http://arxiv.org/pdf/math/9702224v1.pdf, Athanasiadis and Linusson give a bijection between the regions of the Shi ...

**5**

votes

**2**answers

203 views

### 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 ...

**2**

votes

**0**answers

129 views

### 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 ...

**5**

votes

**1**answer

116 views

### 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 ...

**2**

votes

**3**answers

216 views

### 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 ...

**1**

vote

**0**answers

44 views

### 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 ...

**0**

votes

**1**answer

166 views

### 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}), ...

**5**

votes

**0**answers

127 views

### 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 ...

**35**

votes

**0**answers

553 views

### 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 {$L_1,\dots,L_m$} ...

**2**

votes

**0**answers

47 views

### 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 ...

**5**

votes

**2**answers

174 views

### 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 ...

**3**

votes

**2**answers

1k views

### 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 ...

**4**

votes

**3**answers

278 views

### 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 ...

**4**

votes

**1**answer

218 views

### 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 ...

**1**

vote

**0**answers

503 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 ...

**14**

votes

**1**answer

608 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 $\mathbb{R}^n$. ...

**6**

votes

**0**answers

227 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 ...

**4**

votes

**1**answer

242 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}} + ...

**14**

votes

**1**answer

539 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 ...

**12**

votes

**0**answers

546 views

### Pencils with many completely decomposable fibers

Let $F= \frac{G}{H} : \mathbb P^n \to \mathbb P^1$ be a non-constant rational function ($G$ and $H$ homogenous polynomials of the same degree
in $\mathbb C^{n+1})$.
The fiber over $(\lambda:\mu) \in ...