**2**

votes

**0**answers

26 views

### How to prove the existence of the polytope in $\mathbb{R}^d$ with a given number of faces, minimizing the isoperimetric ratio?

This is the isoperimetric type question. We know that in $\mathbb{R}^d$, balls are the sets that minimize the isoperimetric ratio $\frac{S^{d}}{V^{d-1}}$, where $S$ is the surface area and $V$ is the ...

**0**

votes

**0**answers

31 views

### Algebraic independence in normed spaces

A set of $n$ points in $\mathbb{R}^2$ is algebraically independent over $\mathbb{Q}$ if there is no polynomial dependency among the $2n$ coordinates.
A result (Lemma 3.3) from "Globally linked pairs ...

**1**

vote

**1**answer

189 views

### Convex polyhedron and its Gauß-curvature [on hold]

I have asked this question on MathSE and no one could give me an answer. So I'll post my question here.
What I am trying to prove:
A convex polyhedron has positive Gauß-Curvature at every vertex.
...

**1**

vote

**0**answers

91 views

### need clarification of a paper by Fejes Toth

I would like some clarification about the following from Fejes Toth's paper "A stability criterion to the moment theorem"
The setup is:
For each positive integer $n$, let $r(H_n)$ and $R(H_n)$ ...

**4**

votes

**1**answer

109 views

### Motzkin polynomials and enumeration of chord diagrams

On page 12 of the paper Enumeration of chord diagrams on many intervals and their non-orientable analogs" by Alexeev, Andersen, Penner, and Zograf is a list of polynomials which are a refinement of ...

**3**

votes

**1**answer

95 views

### Special case of Erdos Distance Problem in a plane

Erdos in his Distinct distance Problem in a plane conjectured that the minimal number of distinct distance determined by $n$ points in a plane be $g(n)$,
$$g(n) \sim \frac{cn}{\sqrt{\log n}}$$
But ...

**6**

votes

**2**answers

368 views

### Embedding of planar graphs

I've recently come across the following lemma.
Lemma (Valiant): A planar graph $G$ with maximum degree $4$ can be embedded in the plane using $O(|V|)$ area in such a way that its vertices are at ...

**2**

votes

**1**answer

129 views

### Number of lines of symmetry of a set of lattice points

Given some finite $S\subseteq\mathbb R^2$, it is clearly possible for $S$ to have arbitrarily many lines of symmetry. However, it is not very clear if the same is necessarily true for subsets of ...

**3**

votes

**2**answers

290 views

### Geometry, Number Theory and Graph Theory of n-gon, permutation and graph labeling?

Given $n$ and $t$ lengths $ l_i, 1\leq l_1<l_2<\cdots<l_t\leq n-1$, of directed diagonals within an $n$-gon such that $l_1+\cdots+l_t\neq 0 \pmod n)$. Does it exist a directed path within ...

**3**

votes

**1**answer

101 views

### Do random triangulation edge-flips maintain randomness?

Let $S$ be a fixed set of $n$ points in the plane in general position.
Let $T$ be a triangulation
of $S$, (somehow) selected
uniformly at random from all triangulations of $S$.
(There are an ...

**2**

votes

**1**answer

93 views

### Maximal neighbour-full partition of $\{0,1\}^n$

What is the largest complete minor of the $n$-dimensional hypercube? (which we call $k(n)$)
Alternatively, what is the partition of $\{0,1\}^n$ with each set connected and neighboring each other that ...

**8**

votes

**4**answers

252 views

### Diameter of random segment intersection graph?

I have an even number of points $n$ randomly distributed (uniformly) in a disk.
Then the points are randomly connected to form $n/2$ segments, a perfect
matching.
Finally, I form the intersection ...

**10**

votes

**2**answers

276 views

### Double kissing problem

Consider two touching unit balls which will be called central balls. What is the maximum number $k$ of non-overlapping unit balls so that each ball touches as least one of two central balls?
An easy ...

**4**

votes

**1**answer

106 views

### Best polygonal approximation to a polynomial $\pm$ c

Let a planar region $R$ be defined
by the vertical range bounded by
a polynomial $f(x) \pm c$ with $c>0$ a constant,
and with $x$ varying between the smallest and largest
roots of $f(x)$.
For ...

**6**

votes

**5**answers

209 views

### Packing obtuse vectors in $\mathbb{R}^d$

I came across this attractive theorem:
Theorem. In $\mathbb{R}^d$, there can be at most $d+1$ vectors that
form an obtuse angle with one another.
This was proved1 as a corollary of a lemma about ...

**21**

votes

**7**answers

1k views

### What's that shape? Inferring a 3D shape from random shadows

Let $P$ be a bounded, simply connected region of $\mathbb{R}^3$.
$P$ could be a polyhedron, or a smooth shape, or an arbitrary shape;
I'll assume below that $P$ is a (non-degenerate, perhaps ...

**6**

votes

**1**answer

149 views

### Existence of finite set of points in the revolving circles

Let $k$ and $n$ be two fixed integers. Let $C$ denotes the circle with radius $4n$ (in the plane $\mathbb{R}^2$). Suppose $\{C_1,C_2\}$ shows the set of two arbitrary tangent circles with radius $2n$ ...

**1**

vote

**1**answer

188 views

### Ask the name of a combinatorial theorem

It is a classical theorem. For given integer $n \ge 1$, among ${n\choose{n/2}} = 2^{(1-o(1)n)}$ strings in the cube $\{0, 1\}^n$ with weights $n/2$, i.e., $n/2$ indices are 1, there are at least ...

**11**

votes

**2**answers

302 views

### The most number of points that realize only $k$ distinct distances

For $k \ge 1$, let $f_d(k)$ be the largest possible number of points $p_i$
in $\mathbb{R}^d$ that determine at most $k$ distinct (Euclidean) distances,
$\|p_i-p_j\|$.
Example. For points in the plane ...

**3**

votes

**2**answers

162 views

### Examples of toric threefolds

I am looking for examples of smooth projective toric threefolds $\mathbb P_\Delta$ such that the rational polytope $\Delta$ has only pentagonal faces and hexagonal faces.
I quickly searched for ...

**4**

votes

**2**answers

62 views

### Expressing a convex Polytope as a sublevel set of a function

Given an n-dimensional polytope $P$ in $\mathbb R^n$, Given as a convex hull of a finite set of points, $S$ I would like to construct an expliict formula for a function $f\colon \mathbb R^n \to ...

**2**

votes

**0**answers

126 views

### Find the intersection between two convex hulls, in this specific case

We work over $\mathbb{R}^K$. Let $V$ be the set of vectors whose coordinates take values $0$ or $1$, or equivalently the corners of the unit cube $[0,1]^K$.
Let $d:\{0, \ldots, K\} \to \mathbb{R}_+$ ...

**2**

votes

**1**answer

81 views

### Epsilon-approximations of set systems with finite VC-dimension

ECorollary 6.9 in A Guide to NIP theories by Pierre Simon proves the following
Theorem. For every positive integer $k$ and every positive real $\varepsilon$ there is an integer $n=n(k,\epsilon)$ ...

**1**

vote

**1**answer

173 views

### Cardinality of non-integer points in the translation of the Minkowski sum of convex hull.

Let $\operatorname{conv}(a_1,\ldots,a_m)$ denote the convex hull of $\{a_1,\ldots,a_m\}$. Let $\mathbb{Z}_+=\mathbb{N}\cup\{0\}$ and $\mathbb{Q}_+$ denotes the positive (inluding 0) rational numbers. ...

**5**

votes

**1**answer

221 views

### Looking for reference or proof to some facts stated on Anand Pillay's book

In my current work I am using facts 2.1.11 and 2.1.12 from Anand Pillay's book Geometric Stability Theory.
The facts are stated as follows:
Fact 2.1.11. Let $(S,\mbox{cl})$ be a locally ...

**1**

vote

**1**answer

148 views

### Enumerating Lattice points

Let $A \in \mathbb{R}^{d\times d}$ be an invertible matrix. Consider the set
$$P_d := A\mathbb{Z}^d = \{A x| x \in \mathbb{Z}^d \} \subset \mathbb{R}^d$$.
and
$$ Q_d := [-1,1]^d.$$
I am interest in ...

**1**

vote

**0**answers

55 views

### Presentation of the Rybnikov matroid

In this well celebrated work Gregory Rybnikov exhibit an example of two arrangements with the same underlying matroid, but with fundamental groups which are not isomorphic. This is a key ...

**1**

vote

**3**answers

175 views

### Isometric imbedding of finite metric space into standards spaces [duplicate]

Is it true that any metric space consisting of $n$ points can be isometrically imbedded into $n-1$ dimensional Euclidean space? Hyperbolic space?
(For $n=3$ this is true.) If not, what are ...

**23**

votes

**1**answer

441 views

### Expected number of vertices of a hypercube slice — is this new/interesting?

I am a (mostly) amateur mathematician, but my education and work have featured a lot of mathematics, and recently I bumped into a mathematical problem for which I can find no references, and I am ...

**2**

votes

**0**answers

39 views

### A weaker version of Randell Isotopy Theorem

I am studying a problem in hyperplane arrangement theory related to the homotopy type of the complement manifold of a certain class of hyperplane arrangements.
In a well celebrated paper Richard ...

**4**

votes

**2**answers

239 views

### Breaking a rectangle into smaller rectangles with small diagonals

Say I am given a rectangle with dimensions $a \times b$ and an integer $n$. I'd like to break this rectangle into $n$ smaller rectangles $R_i$, and I'd like to make the maximum diagonal of any of ...

**4**

votes

**2**answers

250 views

### What are the applications of Voronoi diagrams in pure mathematics? [closed]

Voronoi diagrams have interesting mathematical properties and applications in algorithms and modeling. But what are its applications in pure mathematics? For example, what theorems can be proved using ...

**7**

votes

**1**answer

208 views

### Convex Polyhedra Scissors Congruence Problem

I am currently writing a geometry paper "Rectifications of Convex Polyhedra" and I am confused to have discovered what appears to be a remarkable discrete geometric fact:
Conjecture: Let $P$ be a ...

**3**

votes

**2**answers

232 views

### Isoperimetric inequality on the Hamming cube

Suppose $X \subseteq \lbrace 0 , 1 \rbrace ^{m}$ such that $|X| \geq 2^{0.8m}$, and $m \geq 2$, then prove that there exists $x,y \in X$ with $||x - y||_{1} \geq m/2$.
My approach to prove this was ...

**19**

votes

**1**answer

298 views

### Maximum height of intersection of triangles

I'd like some advice regarding the following question, which I have been struggling with for long time.
Let's call the shaded region in the below $S_3$. It is the union of three congruent isosceles ...

**8**

votes

**1**answer

317 views

### Orthonormal bases of R^3 with components lying in the golden field

Greg Egan proved an interesting theorem about unit vectors in $\mathbb{R}^3$ whose components actually lie in the 'golden field' $\mathbb{Q}[\sqrt{5}]$. He found it in our studies of twin ...

**46**

votes

**5**answers

2k views

### Do unit quaternions at vertices of a regular 4-simplex, one being 1, generate a free group?

Choose unit quaternions $q_0, q_1, q_2, q_3, q_4$ that form the vertices of a regular 4-simplex in the quaternions. Assume $q_0 = 1$. Let the other four generate a group via quaternion ...

**5**

votes

**1**answer

138 views

### Integer decomposition of dilated integral polytopes

For $n > 0$, let $P$ be an integral polytope, that is, the convex hull in $\mathbb{R}^n$ of points in $\mathbb{Z}^n$. Suppose that $\dim(P) = n$.
Question: Given $d > n + 2$ is it true that
$$ ...

**1**

vote

**0**answers

35 views

### Lattice-isotopic essentialization of arrangements

I'm working on a problem related to
$\textbf{Randell's isotopy theorem}$ for complex hyperplane arrangements. I have a question which seems quite obvious. However, I haven't found a rigorous proof ...

**3**

votes

**1**answer

139 views

### An upper bound on the number of sets of parallel lines covering points in a finite plane?

Let $\mathbb{F}$ be a finite field of characteristic $2$. Let $L_m$ denote the set of lines in $\mathbb{F}^2$ with slope $m\in\mathbb{F}$, that is, all parallel lines of the form $y=mx+b$. Consider a ...

**10**

votes

**0**answers

148 views

### Self-avoiding random walks that always turn

I am wondering if the statistics of self-avoiding random lattice-walks
on $\mathbb{Z}^2$
that turn left or right at each step (i.e., they cannot continue the
direction of the preceding step) have been ...

**25**

votes

**0**answers

614 views

### 3-colorings of the unit distance graph of $\Bbb R^3$

Let $\Gamma$ be the unit distance graph of $\Bbb R^3$: points $(x,y)$ form an edge if $|x,y|=1$.
Let $(A,B,C,D)$ be a unit side rhombus in the plane, with a transcendental diagonal, e.g. $A = ...

**4**

votes

**0**answers

78 views

### How many subspaces are generated by three or more subspaces in a Hilbert space?

In the book of G. D. Birkhoff "lattice theory", it is mentioned that there are 28 subspaces that can be obtained from three subspaces in general position in a Hilbert space (using intersections and ...

**1**

vote

**1**answer

128 views

### Chern classes of three (two) dimensional complex vector bundles

Let $M$ be a manifold.
Let $F(M,3)=\{(m_1,m_2,m_3)\mid m_1, m_2, m_3\in M, m_i\neq m_j, \text{ for any } i\neq j\}$.
Let $S_3$ be the symmetric group of order $3$.
Let $S_3$ act on $F(M,3)$ by ...

**8**

votes

**1**answer

82 views

### A variant of Nelson-Hadwiger Problem on the chromatic number of the plane

The famous Nelson-Hadwiger problem asks about the chromatic number of the graph $G$, with the vertex set $V(G)={\mathbb R}^2$ where $a_1=(x_1,y_1), a_2=(x_2,y_2) \in V(G) \ $ form an edge iff ...

**3**

votes

**2**answers

591 views

### What's the name of this geometric mathematical modeling problem?

There is a right angle corner with width 1 in both directions. One wants to find the largest area shape which can pass through this corner.
I know that this is a famous problem, but what is it called?
...

**2**

votes

**0**answers

88 views

### Computing Voronoi poles in $\mathbb{R}^d$ (the farthest points within each cell)

Say I have a Voronoi diagram of some points $p_1,\dots,p_n\in\mathbb{R}^d$, which tesselates $\mathbb{R}^d$ into cells $V_1,\dots,V_n$. Within each cell $V_i$, the pole is defined as the vertex of ...

**5**

votes

**1**answer

130 views

### Upperbounding the number of regions induced by a set of unit disks

Given a set $D$ of $n$ same radius disks, embedded in the plane, their arrangement induces a number $k$ of connected regions in $\mathbb{R}^2 \setminus \cup_{d \in D}$ .
I am interested in an upper ...

**1**

vote

**1**answer

41 views

### number of affine pieces of linear interpolation of convex functions in high dimension

Consider a convex function $f$ defined on a $d$-dimensional hypercube $[0,1]^d$. Now for fixed $m \in \mathbb{N}$, consider the grids $\mathcal{G}_m=\{(i_1/m,\cdots,i_d/m)\}$ where ...

**3**

votes

**3**answers

333 views

### Aperiodic graphs

The concepts of being non-periodic and aperiodic for tilings have obvious versions for connected graphs with a countable set of vertices and a finite number of edges meeting at each vertex. A graph ...