Discrete geometry includes the study of covering, illumination, packing, convex bodies, convex polytopes, and other metric geometry.

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Dirichlet's approximation theorem (simultaneous version): proof via Minkowski's theorem

There is a proof of the Dirichlet's approximation theorem based on Minkowski's theorem. The proof is given on wikipedia (http://en.wikipedia.org/wiki/Dirichlet%27s_approximation_theorem) and it is ...
3
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1answer
44 views

Triangulation of hypercubes into simplices

A square can be divided into two triangles. A 3-dimensional cube can be divided into 6 tetrahedrons. Into what number of simplices an n-dimensional hypercube can be divided? (For example, a ...
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1answer
17 views

Prove that a planar bipartite graph on n nodes has at most 2n−4 edges.

I know that we have to use Euler's formula ( v−e+f=2) but I don't understand how f = e/2.
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1answer
22 views

How Many Triangles are Created by n Lines in the Plane?

Suppose we are given n lines in the plane in "general position", which in the present case we define to mean the following: A. no 2 lines are parallel or identical B. no 3 lines have common ...
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2answers
149 views

Partitioning the plane into three sets each intersecting the vertices of every square with side 1?

Q1. Is it possible to partition the plane into three sets such that each of them contains at least one vertex of every square with side 1 ? (I mean all squares of side-length 1, not just those with ...
2
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1answer
47 views

Euler characteristic of closed surface

Assume that you have a closed surface that can be covered by finitely many triangles. Then $K(p)= 6-val(P)$ where P is a vertex and $val(P)$ the number of edges that lead to this vertex. Now, I am ...
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1answer
37 views

How to define open and closed functions whose domain or range is a discrete metric space?

I encountered that a function is open or closed in my analysis book [Herbert Amann, 2005], and it illustrates it in this way: A function $f: X \xrightarrow{} Y$ between metric spaces $(X,d)$ and ...
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3answers
29 views

Is there a theorem or axiom which shows that permutations of step sequences through a lattice graph result in the same destination?

I have been searching for a theorem, lemma, or even an axiom which shows that the permutations of a step sequence in Taxicab Geometry result in the same destination in such a lattice graph. To ...
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6answers
852 views

Prove Existence of a Circle

There are two circles with radius $1$, $c_{A}$ and ${c}_{B}$. They intersect at two points $U$ and $V$. $A$ and $B$ are two regular $n$-gons such that $n > 3$, which are inscribed into $c_{A}$ and ...
0
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1answer
26 views

Prove that a convex $d$-polytope has at least $d+1$ facets

This seems trivial but I can't come up with a formal proof. I think there should be a way to do this inductively but I can't figure out how$\ldots$ Any help much appreciated
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22 views

Discrete Geometry (Polytopes)

I have to try to prove the following: Let $V = {v_1,...,v_n} \subset R^d$ be a point configuration affinely spanning $R^d$ (i.e., $aff(V) = R^d)$. Let H be the collection of hyperplanes spanned by ...
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1answer
19 views

Proof: Minkowski sum polytope implies A and B polytopes

Suppose $A$ and $B$ are convex sets and their Minkowski sum $A+B$ is a polytope. How do you prove that $A$ and $B$ are polytopes as well?
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47 views

Are there any four-dimensional shapes in the whole wide world?

I've looked up images of a 4-D (four-dimensional) shape and they looked like there are built by using regular 3-D (three-dimensional) shapes using a regular 3-D shape connected to another 3-D shape ...
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4answers
6k views

Why is a circle in a plane surrounded by 6 other circles

When you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other number? I'm ...
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1answer
30 views

Relative Interiors of polyhedra

***Source article: Magnanti, T. L., & Wong, R. T. (1981). Accelerating Benders decomposition: Algorithmic enhancement and model selection criteria. Operations Research, 29(3), 464-484
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1answer
33 views

Does a unipotent transformation preserve covolume?

Let $S\subset \mathbb{R}^d$ and define $v(S)$ to be the volume of the set $(S+\mathbb{Z}^d)\cap [0,1]^d$ (where $[0,1]^d$ is the unit cube $[0,1]\times [0,1] \times...\times [0,1]$). Let $T: ...
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0answers
16 views

dimension of Weber set and selectope (as a operator)

Let $\Omega$ be a finite set of players. For a selector $\alpha:(2^{\Omega}-\{\emptyset\})\rightarrow\Omega$, we define a marginal value operator as a linear operator $m^{\alpha}$ ...
3
votes
2answers
172 views

crossing number question

Prove that there exists constant k such that, for all $5v < e$ there is a subgraph of the complete graph of $v$ vertics with crossing number less or equal than $ k e^3/v^2$. Any hints for a way to ...
3
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0answers
56 views

Number of lines needed to pass through every region of a map

The webpage http://what-if.xkcd.com/113 explores the fewest number of lines needed so that every state in the US has at least one line going through it. (actuallly great circles on a sphere) Can you ...
6
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3answers
265 views

On coverings of the complex sphere

Here, everything takes place in $\mathbb{C}^d$ for some $d$, and the sphere $\mathcal{S} = \{\mathbf{x}\in\mathbb{C}^d:\|\mathbf{x}\| = 1\}$. Given $\delta > 0$, consider a collection of vectors ...
0
votes
1answer
22 views

dual set of the dual set

Let $X\subseteq\mathbb{R}^d$ and let $X^*$ be it's dual set i.e. $X^*=\{y\in\mathbb{R}^d| <x,y>\leq 1$ for every $x\in X\}$. How to prove that $(X^*)^*=\overline{conv(X\cup\{0\})}$? I know that ...
3
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1answer
111 views

Is every finite group of isometries a subgroup of a finite reflection group?

Is every finite group of isometries in $d$-dimensional Euclidean space a subgroup of a finite group generated by reflections? By "reflection" I mean reflection in a hyperplane: the isometry fixing a ...
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2answers
77 views

Calculating volume of spherical wedge from parallelepiped corner

I am interested in calculating the volume of the intersection of a sphere of radius 1/2 with the corner of a parallelepiped where the angles between each edge is $\pi/3$ and has unit edge length; we ...
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2answers
55 views

Simplest graph that is not a segment intersection graph

Given a finite collection $S=\{s_1,s_2,\ldots,s_n\}$ of straight-line segments in the plane, their intersection graph $G(S)$ is a graph that contains a vertex $v_i$ for each segment $s_i\in S$, and an ...
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0answers
18 views

Existence objective function given optimality regions

Let $I$ and $X$ be finite, nonempty sets, and denote by $\Delta(X)$ the set of probability measures on $(X,2^X)$. Suppose that for each $i \in I$, we are given a subset $M_i \subseteq \Delta(X)$ of ...
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0answers
31 views

Constrained optimization using a cutting plane on a tetrahedron

Consider the figure below where $(a,b,c,d)$ is a tetrahedron and $p=(1-t)a+tb$ is a point on the $ab$ segment. If $n_a$ and $n_b$ are two unit vectors associated with $a$ and $b$, respectively, then ...
3
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1answer
150 views

Tetromino Proof

Prove that an 8 x 8 board cannot be covered by 15 L-tetrominos and one square tetromino (an L-tetromino is a plane figure shown below, constructed from four unit squares arranged in the form of L; a ...
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1answer
151 views

Schlegel diagram of polytope

I'm currently studying polytopes and using the book Lectures in Geometric Combinatorics by Thomas. When I come to Schlegel diagrams, I do not quite understand how to determine whether a Schlegel ...
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0answers
40 views

Krull dimension and Hilbert polynomial of graded rings

Let $P \subseteq \mathbb{R}^n$ be a polytope with integral vertices, and $Q := \mathbb{R}_+ (P \times \{1\}) \cap \mathbb{Z}^{n+1}$ be the monoid of all lattice points of the cone generated by $P$ in ...
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1answer
156 views

The lattice points of a f.g. rational cone form a f.g. monoid.

In their book "Polytopes, Rings, and K-Theory" Bruns and Gubeladze sketch an alternative approach to Gordan's Lemma, which is stated in the headline (f.g. = finitely generated). I don't understand the ...
2
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1answer
33 views

Smallest triangle in a convex polygon triangulation

I have been working on this problem for quite a while and it seems necessary to prove or disprove this particular problem. Suppose $T$ is the set of all possible triangles made from the vertices of a ...
3
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1answer
192 views

Algebraic proof of Ehrhart's theorem

Let $P \subset \mathbb{R}^d$ be a $d$-dimensional polytope, where all vertices lie on integral coordinates, and let $L(P,n)$ denote the number of integral lattice points contained in the scaled ...
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0answers
47 views

A continuous centerpoint of a convex spherical polygon

In discrete geometry, a centerpoint $c$ of a discrete set $S$ of $n$ points in the plane is such that any half plane containing $c$ contains (roughly) $n/3$ points of $S$. (Such a centerpoint always ...
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2answers
127 views

Regular Pentagon is the Unique Largest Two-Distance Set in the Plane

A two-distance set is a collection of points for which only two distinct distances appear among pairs of points. (That is, the distance between any pair of points is either $x$ or $y$, and these ...
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1answer
35 views

Small remarkable matroids

I'm working on a problem involving matroids $M=(E,\mathfrak{C})$ (here $E$ is the ground set, $\mathfrak{C}$ the set of circuits) with a "small" ground set $E,$ in the sense that $\sharp(E)\leq7$ I ...
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0answers
34 views

Ehrhart polynomial of lattice tetrahedrons in $\Bbb{R}^4$

Let $\lbrace v_1 , v_2, v_3 , v_4 \rbrace \subset \Bbb{Z}^4$ be linearly independent, and denote by $P$ the convex hull of this set. Now, $P$ is a 3-polytope residing in four-dimensional space. What's ...
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1answer
24 views

Wrong formulation of Helly's theorem

In Lectures on Discrete Geometry, Matousek writes (p.11) (excerpt here): It is very tempting and quite usual to formulate Helly's theorem as follows: "If every $d+1$ among $n$ convex sets in ...
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1answer
65 views

Carathéodory's convex hulls theorem and Radon partitions

Wikipedia's article about Radon's theorem and its related states: Carathéodory's theorem states that any point in the convex hull of some set of points is also within the convex hull of a subset ...
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1answer
54 views

Topological subspace in $(S^{1})^{n}$

Studying the set of solutions of a particular linear system associated to a matroid, I notice that is it possibile to determine the topology of the quotient and identify it as a subtorus of ...
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1answer
36 views

2D discrete Jordan curve theorem: what about the “boundary points”?

Let us say we have a polygon $P$ in $\mathbb{R}^2$, with edges in the set $E$ (the boundary), and vertices in the set $V$. Let us say we have a point $Q$ such that $Q$ lies on one of the edges in $E$. ...
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0answers
50 views

Regular polygons constructed inside regular polygons

Let $P$ be a regular $n$-gon, and erect on each edge toward the inside a regular $k$-gon, with edge lengths matching. See the example below for $n=12$ and $k=3,\ldots,11$.       Two ...
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1answer
42 views

Volume of a parallelopiped

Suppose $\Lambda$ is a lattice in $\mathbb{R}^n$ of rank $r$ and $\mathbf{b}_1, ..., \mathbf{b}_r \subseteq \mathbb{R}^n$ its basis. I know that if we pick any orthonormal vectors $\mathbf{e}_{r+1}, ...
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2answers
156 views

Squaring the plane, with consecutive integer squares. And a related arrangement

Q1. I was fiddling around with squaring-the-square type algebraic maths, and came up with a family of arrangements of $n^2$ squares, with sides $1, 2, 3\ldots n^2$ ($n$ odd). Which seems like it would ...
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0answers
34 views

Squaring the plane with consecutive integer squares. And a related arrangement. [duplicate]

Q1. I was fiddling around with squaring-the-square type algebraic maths, and came up with a family of arrangements of $n^2 $squares, with sides $1,2\ldots n^2$ (n odd). Which seems like it would work ...
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12 views

About the logistic map.

I need guide line about it I also wanted to know how it will appear in graph if we use mathematica or some other software for this.
2
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1answer
94 views

Question related to a determinant of lattice

There is this equation related to the determinant of lattice and I have been stuck on it for a little while. I would greatly appreciate if someone could explain to me how to prove it! Let $\Lambda$ ...
0
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1answer
35 views

Is mean pairwise distance a metric over subsets of a metric space.

Specifically, I am looking at finite subsets of a set that is a discrete metric space under Jaccard Distance. I'm having trouble proving the triangle inequality or coming up with a counterexample. ...
0
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1answer
31 views

Why does a 2-colourable simple graph with n nodes have no more than $(n^2/4)$ arcs?

Why does a 2-colourable simple graph with n nodes have no more than $(n^2/4)$ arcs? I would really appreciate for any kind of explanations.
3
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1answer
32 views

Graph of polytope and hyperplane

Suppose that $P$ is a compact and convex polytope in $R^d$ and let $G$ be the graph of $P$ ($V(G)$ are the vertices of $P$ and $E(G)$ are the $1$-dimensional faces - for example polyedral graphs are ...
2
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1answer
119 views

Geometry textbook

I am planning to take a graduate Geometry course next semester. The preliminary syllabus does not specify any textbook but has the following descriptions: Catalog Course Description: This course ...