Tagged Questions

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

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2
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0answers
20 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 ? Q2. Let $n$ be the largest integer such that the plane ...
0
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1answer
19 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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0answers
16 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 ...
0
votes
1answer
16 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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0answers
43 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 ...
1
vote
1answer
23 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
2
votes
1answer
32 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: ...
2
votes
0answers
15 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 $m^{\alpha}$ ...
25
votes
0answers
688 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 ...
3
votes
0answers
53 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
votes
3answers
262 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
17 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
votes
1answer
92 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 ...
1
vote
2answers
68 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 ...
6
votes
2answers
54 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 ...
0
votes
0answers
30 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 ...
0
votes
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 ...
1
vote
0answers
29 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 ...
2
votes
1answer
29 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 ...
5
votes
1answer
148 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 ...
0
votes
0answers
42 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 ...
2
votes
0answers
32 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 ...
0
votes
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 ...
1
vote
1answer
21 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 ...
1
vote
1answer
57 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 ...
1
vote
1answer
52 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 ...
3
votes
1answer
190 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 ...
0
votes
1answer
33 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 $E$. ...
5
votes
0answers
47 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 ...
1
vote
1answer
41 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}, ...
2
votes
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 ...
6
votes
2answers
151 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 ...
0
votes
0answers
11 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
votes
1answer
90 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
votes
1answer
31 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
votes
1answer
30 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.
2
votes
1answer
105 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 ...
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vote
0answers
99 views

Invertibility of NxN nonnegative matrix with diagonally dominant elements

I have a NxN nonnegative matrix where the diagonal element of any row i is greater than the off diagonal elements, 1 > aii > aij ≥ 0 for j not equal to i . This not a diagonally dominant matrix as it ...
0
votes
0answers
36 views

A configuration of 4 points on the plane that minimizes the variance of the distances between the points

This is connected to my earlier question: Embedding finite (discrete) metric spaces to Eulidean space as isometrically as possible but now I'm only considering the smallest (non-trivial) case. We ...
2
votes
1answer
68 views

Embedding finite (discrete) metric spaces to Eulidean space as isometrically as possible

Let $X = \{1, 2, 3, ..., k\}$ with the discrete metric (distance is 1 for every pair of points). How can this be embedded into $\mathbb{R}^n$ (with the usual metric) such that the embedding would be ...
3
votes
0answers
90 views

The Erdős-Szekeres problem on points in convex position

The Erdős-Szekeres problem on points in convex position and its proof using Ramsey theorem are well know. The problem goes like this: For every natural number $k$ there exists a number $n(k)$ such ...
3
votes
1answer
163 views

Poisson point process (PPP) and Voronoi cells

Say we have a homogeneous PPP with rate $\lambda$ in the 2-D plane $\mathbb R^2$. In one realization of the PPP we get the points $\phi=\{x_1,x_2,...,x_i,...\}$. Now we generate the Voronoi cells ...
1
vote
1answer
105 views

Cover the n-sphere with sub-hemispherical caps

Original Question (answered): Define a cap (x,Phi) to be the set of all points of the sphere that are within an angle Phi of the point x. $ 0 \le \phi < \frac{\pi}{2} $. (define the angle ...
3
votes
1answer
26 views

Cyclic polytope of dimension 4

I don't quite know how to count the number of $k$-dimensional faces of a $4$-dimensional cyclic polytope (http://en.wikipedia.org/wiki/Cyclic_polytope) without using the standard formula. Any advice? ...
3
votes
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 ...
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vote
2answers
42 views

Proof for the length of the shortest 4-connected path and 8-connected path on a chessboard

I have a chessboard with a square marked with A as in the following figure: $$\begin{array}{|c|c|c|} \hline 8&1&2\\ \hline 7&A&3\\ \hline 6&5&4\\ \hline \end{array}$$ The ...
2
votes
0answers
52 views

Generalization of Minkowski's theorem

I would like to prove a generalized version of the Minkowski's theorem, but I don't quite know how to do it. Here is what I would like to prove: Let $X\subset \mathbb{R}^d$ is convex, symmetric ...
3
votes
1answer
140 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 ...
2
votes
1answer
55 views

Reference for important results in linkage theory and their proofs

Are there books or lecture notes that comprehensively introduce the (geometric/topological) theory of mechanical linkages, as well as important results and their proofs? For instance, Kempe's ...
4
votes
0answers
116 views

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 ...