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

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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 ...
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2answers
45 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
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0answers
55 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
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 ...
2
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1answer
58 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
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1answer
128 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 ...
6
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0answers
307 views

Surface integral for a scalar function defined on a discrete surface

Imagine a polyhedral, discrete surface embedded in $\mathbb{R}^3$. Its faces are all triangles. For each vertex, one can compute the discrete mean and Gaussian curvatures and evaluate the sum of ...
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1answer
82 views

If we spot n dots on a sphere (radius r), there exists two dots whose distance is under d

Is there any formula about $n$, $r$, $d$ at the question?
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1answer
93 views

Question related to Desargues' Theorem

The diagram below is one way of drawing two triangles ($\Delta PQR,\ \Delta P'Q'R'$) perspective from a point ($O$), with pairs of corresponding sides meeting at $D, E, F$ as in Desargues' Theorem ...
11
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1answer
443 views

Circles on the plane such that every line intersects at least one of them but no line intersects more that 100 of them

I have a serious problem with this problem: Is it possible to Draw circles on the plane such that every line intersects at least one of them but no line intersects more that 100 of them !? Any help ...
0
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1answer
106 views

Flip graph of point set [closed]

Is the flip graph of every point set in $\mathbb R^3$ connected? If not, is there a set with an isolated node? Def: For a point set $S$, the flip graph of $S$ is a graph whose nodes are the set of ...
2
votes
1answer
45 views

Filling Ratio of Unit Sphere

Consider the unit sphere $S^n$ in ${\bf R}^{n+1}$. Consider $S(r)$, a union of $r$-balls in $S^n$ which is disjoint and that $S(r)$ has maximum area. Then define $$ c_n\doteq ...
2
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0answers
96 views

Intersection of line with discrete hypercubes in n-dimensional space

I am looking for a method to determine the hypercubes that intersect a line between two points in a high dimensional space. I think what I want is the supercover of a line in high dimensional space. ...
2
votes
1answer
89 views

halving lines through the centroid of a cyclic polygon

Let $A_1, A_2,\ldots, A_{2n}$ be $2n$ points on a circle centered at $O$ with the additional property that the centroid of this set of points coincides with $O$. In other words, the sum of the vectors ...
6
votes
0answers
110 views

Covered 10x10 rectangle with L-shapes trominos

We have given L-shaped trominos and a square of size 10x10. Give a nice proof, that 18 L-trominos is the minimal number with which the square can be covered such that it is impossible to insert one ...
5
votes
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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0answers
35 views

Weights for degree ordering

Let $x_1,x_2,x_3$ be indeterminates. Fix an integer $k\geq 3$. Consider the set $M$ of all monomials of the form $x_1^{i_1}.x_2^{i_2}.x_3^{i_3}$ where each $i_j\in \mathbb{N}$ with $i_j\geq 1$ and ...
7
votes
1answer
125 views

Helly's Theorem for Biconvex Sets

Helly's Theorem states the following. Suppose that $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq ...
25
votes
4answers
544 views

square cake with raisins

Alice bakes a square cake, with $n$ raisins (= points). Bob cuts $p$ square pieces. They are axis-aligned, interior-disjoint, and each piece must contain at least $2$ raisins. Note that a single ...
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0answers
101 views

discrete harmonic extension (an exercise of Grimmett's “probability on graphs”)

I'm struggling with exercise 1.3 in Grimmett's book "probability on graphs". Take $G = (V,E)$ a finite connected graph with given positive conductances $(w_e)_{e \in E}$, and let $(x_v)_{v \in V}$ be ...
2
votes
1answer
49 views

Simplex with edges of length at least s having smallest circumradius

Is it true that of all $k$-simplices with edge lengths greater than or equal to some parameter $s$, the regular simplex with edge lengths $s$ has the smallest circumradius? Please supply a proof or ...
3
votes
1answer
58 views

Simple geometry problem on distribution of points in a plane

Consider 6 distinct points in a plane. Let $m$ and $M$ be the minimum and maximum distances between any pair of points. Show that $M/m \ge \sqrt3$. I am more interested in arrangement of these points ...
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2answers
96 views

Approximating Euclidean geometry, restricted to $\mathbb{Q}$

I'm having trouble putting this into a fully coherent question, so I'll give the broad question, then a few bullet points to give you a better idea of what I'm asking. I'm looking for a line of ...
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0answers
55 views

Weighted graphs to minimise the set of distances

$c_{0}$ to $c_{3}$ are given points of the graph and the corresponding weights are $W_{1}$ to $W_{3}$. The objective is to locate $p_{1}$ and $p_{2}$ to minimise the distances $d_{0}+d_{1}$ and ...
2
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2answers
104 views

Discretizing continuous surfaces into semi-regular polygons

I am aware that there have been many works on the problem of discretizing a surface into polygons, however, I wonder if in any work the problem of doing so to get polygons with edges of the same ...
4
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3answers
249 views

Dissecting an equilateral triangle into equilateral triangles of pairwise different sizes

It is know that a square can be dissected into other square such that no two of the squares have the same size. This is the simplest dissection of that kind: Is it also possible to dissect an ...
0
votes
1answer
40 views

Number of faces of $n$ congruent disks

If I have $n$ disks, all of the same radius, how many faces (i.e. maximally connected regions) can the induced arrangement have? For example for 3 disks, it could have 7 bounded faces, but what is ...
2
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0answers
40 views

Fractional Helly for more than one piercing

Fractional Helly Theorem says the following: For every $0<\alpha\leq 1$ there exists $\beta = \beta(d, \alpha)$ with the following property. Let $C_1 , C_2 , ..., C_n$ be convex sets in $R^d$, $n ...
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1answer
139 views

example that shows that the edge chromatic number may be larger than the maximal degree

What is an example that shows that the edge chromatic number may be larger than the maximal degree ∆ ≤ X’(G)
0
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1answer
31 views

Elementary doubt about derivations on Jacobians

Hi I have a small doubt on Discrete Geometry, more specifically the derivation of Jacobian. Say we have a function $x:\mathbb{R}^2\supseteq U \rightarrow \mathbb{R}^3 : (u,v) \mapsto x(u,v)$ Also ...
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0answers
71 views

Combinatorial Laplacian Spectrum

The spectrum of the combinatorial laplacian is well understood for a square lattice. What about for other lattices? In particular: let $ f: \mathbb{Z}^2 \rightarrow \mathbb{R} $. The usual ...
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0answers
47 views

Optimal way to place a given number of points in a region?

Let $A\subset\mathbb{R}²$ and $n\in\mathbb{N}$ be a given natural number. How to find a finite subset of $A$, $P=${$p_1,...,p_n$} such that $\int_A f_P(x)$ is minimum, where $f_P(x) = ...
7
votes
3answers
267 views

Name this polytope

I was wondering what people call a certain type of shape. It is the shape formed by an orthogonal projection of a hypercube along one of its longest diagonals. In other words, fill in the missing ...
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vote
3answers
538 views

Proving by induction that an equilateral triangle will always be divided into (n+1)^2 small triangles?

I'm working on a proof that looks like this: Let $n$ be a positive integer. Given an equilateral triangle, place $n$ points on each side, dividing the side into $n+1$ equal segments. Use the ...
1
vote
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 ...
6
votes
2answers
135 views

Voronoi Diagrams Proof

I am having a real problem with this proof about voronoi diagrams: Prove that $V(p_i)$ (i.e., the cell of $\operatorname{Vor}(P)$ which corresponds to $p_i$) is unbounded if and only if $p_i$ is on ...
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0answers
69 views

Upper bounds on rate of q-ary codes

Among the many upper bounds for families of codes in $\mathbb F _2 ^n$, the best known bound is the one by McEliece, Rodemich, Rumsey and Welch (MRRW) which states that the rate $R(\delta)$ ...
4
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0answers
178 views

Tilings of the plane

There are many possible tilings (or tesselations) of the plane: periodic ones by a - necessarily - finite number of prototiles (e.g. regular tilings) aperiodic ones by a finite number of prototiles ...
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0answers
102 views

Twisted tori: discrete and continuous

Taking the advice of Mariano Suárez-Alvarez, I moved this question from MO to MSE: Motivation Let me introduce twisted (discrete) tori: Consider the Cartesian graph product $\mathcal{C}_n = C_n ...
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1answer
111 views

Unit Distance Problem Formulated as Point-Circle Incidence Problem

The unit distance problem in the plane asks for the maximum number $U(n)$ of unit distances which can be obtained by $n$ points. For $k$ unit circles and $m$ points in the plane, $I(k,m)$ counts the ...
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0answers
69 views

Brouwer’s fixed point theorem ⇒ Sperner’s lemma [duplicate]

Possible Duplicate: Equivalence of Brouwers fixed point theorem and Sperner's lemma Does anyone know a combinatorial proof of the implication from Brouwer’s fixed point theorem to ...
2
votes
2answers
173 views

Triangular grid with 4 edges per vertex

I am trying to create a triangular grid/mesh for a rectangular domain in $\mathbb{R}^2$ with the property that each vertex is shared by (at most) four edges. Is this possible to accomplish?
2
votes
1answer
174 views

Brouwer's fixed point theorem implies Sperner's lemma

I'm looking for a proof that Brouwer's fixed point theorem implies Sperner's lemma. The proof should be understandable by an undergraduate. Thanks in advance!
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0answers
222 views

Sphere Covering Problem

Is it possible that one can cover a sphere with 19 equal spherical caps of 30 degrees(i.e. angular radius is 30 degrees)? A table of Neil Sloane suggests it is impossible, but I want to know if anyone ...
3
votes
1answer
127 views

Can all convex polytopes be realized with vertices on surface of convex body?

Each convex polytope $P$ has a combinatorial type, its so-called face lattice. This lattice is just the poset of all faces of $P$ ordered by inclusion. Given one realization of such a combinatorial ...
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0answers
103 views

“Round” regions on surface of convex polytope

A convex $d$-polytope $P$ is the convex hull of finitely many points. Given such a polytope with $n \gg d$ vertices, I would like to prove that its surface has to be "round" in some region. Let me ...
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0answers
51 views

computing centerpoint using linear programming

I was redirected from math.stackoverflow. So I am trying to teach myself discrete geometry and have started with the centerpoint problem. Could someone please help me understand computing ...
4
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3answers
297 views

Average degree of convex hull vertices in a Delaunay triangulation

Let $P \subset \mathbb{R}^2$. The boundary of $DT(P)$, the Delaunay triangulation of the point set $P$, is $conv(P)$. It is also known that the average degree of the vertices of $DT(P)$ is $\lt 6$. ...
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1answer
94 views

A lemma regarding cones covering $\mathbb{R}^d$

Added: Pointers to some references with the same conclusion as the wolloing lemma may be helpful to understand it, and are appreciated. In A Probabilistic Theory of Pattern Recognition By Luc ...
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0answers
347 views

Parallel transport in discrete differential geometry - programming a game

I would like to get a better intuitive grip on how parallel transport works. I once saw a video a German guy made with a little car having a gyroscope. That car was dragged on a big beach ball and the ...