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

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7
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3answers
288 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 ...
1
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3answers
629 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
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1answer
172 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
162 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 ...
3
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0answers
78 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
191 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 ...
2
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0answers
106 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 ...
1
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1answer
116 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
184 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
181 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!
4
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0answers
262 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
136 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 ...
3
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0answers
107 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 ...
4
votes
3answers
322 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 $\lt 6$. ...
0
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1answer
97 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 ...
3
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0answers
397 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 ...
29
votes
1answer
1k views

Dividing a square into equal-area rectangles

How many ways are there to tile an $n\times n$ square with exactly $n$ rectangles, each of which has integer sides and area $n$? The sequence $C(n)$ begins 1, 2, 2, 9, 2, 46, 2, 250, 37. Clearly ...
3
votes
3answers
232 views

Help me name or find the existing name for this geometric concept!

This may have a proper name, if so - let's discuss. If not, let's name it. This is for a web application in C#, so whatever we call it I will start naming as such in my code. I'm taking GPS data as a ...
4
votes
1answer
128 views

Convex hull of $n$-gon and $m$-gon

Suppose you have a convex $n$-gon, and a convex $m$-gon, in the plane. Take the convex hull of the $n+m$ vertices. How many combinatorially distinct hulls can be obtained, where two hulls are ...
6
votes
1answer
686 views

Intersection of squares/cubes/hypercubes.

One can form a polygon of $4 n$ sides by intersecting $n$ congruent squares (treated as closed sets, i.e., filled squares):          Q1. For which of the ...
3
votes
0answers
75 views

Polytopes-Discrete Geometry

Can someone help me solve the following question please? Let v be a vertex of a d-polytope P such that $ 0 \in intP $ . Prove that $ P^{*} \cap \{ y \in \mathbb{R}^d \mid\left < y, ...
1
vote
1answer
122 views

Restricted Cube Packing

I want to pack n cubes in 3-space under the following 3 restrictions: 1) At each vertex only 2 cubes may touch 2) No two cubes may share an edge 3) No two cubes share any subface 2,3 just mean ...
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1answer
98 views

Maximum number of points with a fixed minimum distance in a $d$-dimensional ball

Let $c \leq r$ be real numbers greater than $0$, $d \in \mathbb{N}$ and $B_r(0) = \lbrace x \in \mathbb{R}^d \mid \Vert x \Vert \leq r \rbrace$, the ball with radius $r$ at point $0$ ($\Vert \cdot ...
3
votes
2answers
175 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 ...
9
votes
4answers
1k views

Every polygon has an interior diagonal

How does one prove that in every polygon (with at least 4 sides, not necessarily convex), that it is possible to draw a segment from one vertex to another that lies entirely inside the polygon. In ...
4
votes
3answers
129 views

What is the name of this property?

If there are 3 intervals, such that any 2 of them intersect, then all 3 of them intersect. For any 4 disks, if any 3 of them have a non empty intersection, then all 4 of them have a common ...
5
votes
3answers
242 views

Number of point subsets that can be covered by a disk

Given $n$ distinct points in the (real) plane, how many distinct non-empty subsets of these points can be covered by some (closed) disk? I conjecture that if no three points are collinear and no four ...
3
votes
2answers
263 views

Find center of circle of radius $r$ that overlaps exactly $\lfloor \pi r^2 \rceil$ points of the integer grid

Can a circle of a given radius $r$ always be placed (in $\mathbb{R}^2$) such that the number of points with integer coordinates inside the circle is equal to the nearest integer of the circle's area? ...
3
votes
1answer
163 views

Intersection of Disks

If I have a disk $d$ where each point of the disk is contained in at least $k$ other disks, then at least how many other disks does $d$ intersect? Given, that all the disks (including $d$) have the ...
0
votes
1answer
77 views

$n$ points in disk, determine number of close distances

If I have a disk of radius $r$, and $n$ points inside this disk, I'm interested in the minimum number of distances $\leq r$ between the points, when the minimum is taken over all $n$ point sets in ...
2
votes
1answer
85 views

How to calculate number of lumps of a 1D discrete point distribution?

I would like to calculate the number of lumps of a given set of points. Defining "number of lumps" as "the number of groups with points at distance 1" Supose we have a discrete 1D space in this ...
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2answers
1k views

Mean distance between N equidistributed points in a circle

I would like to calculate the mean distance depending on circle shape points, This is a mean calculating all posible distances between any two points for N=2, line, there is only 1 distance. for ...
8
votes
1answer
366 views

“Center-of-Mass” of lattice polygons (generalization of Pick's theorem)

Call a polygon with integer coordinates (in the Euclidean plane) a 'lattice polygon'. Pick's Theorem allows you to efficiently compute the number of lattice points inside this polygon given just its ...
4
votes
1answer
387 views

Which internal angles can a lattice polygon have?

I am wondering if for a lattice polygon an internal angle can take any value? If no which ones not and why? I guess there will be some restrictions due to the discrete nature of the grid but I am ...
2
votes
2answers
207 views

Cover n points with n disjoint unit disks

This is a problem I saw on Peter Winkler's column on communication of the ACM(might be under a pay wall). It is open. What is the largest $n$, such that you can always cover a given set of $n$ points ...
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0answers
88 views

Lipschitz constant of the Laplace-Beltrami operator

I'm reading a paper on discrete differential geometry: Meyer et.al. They define the Laplace-Beltrami operator at a point $P$ by $$\vec{K}(p) = 2k_H(P)\vec{n(P)}$$ where $\vec{n}(p)$ is the normal ...
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1answer
85 views

Cutting a d-simplex

Why is it possible to get any possible subset of nodes of a d+1 simplex in IR^d using halfspaces?
8
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4answers
1k views

Definition of simplex

From Wikipedia: an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. I was wondering if the definition is equivalent to say a simplex is synonym of a ...
4
votes
1answer
586 views

Applications of Morse theory

Background The use of tools from algebraic topology to study simplicial complexes coming from point cloud data has been thoroughly discussed in the papers of Carlsson, Zomorodian, Ghrist, ...
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0answers
103 views

Complexity of Counting the number of inducing $n$-gons

Definition: A $n$-gon is simple if it has no self intersection and is in general position if no pair of its edges are parallel. It is clear that by extending the edges of each simple $n$-gon in ...
2
votes
1answer
279 views

How close are star-convex sets to convex sets?

What interesting properties of convex sets are retained by star-convex sets?
11
votes
1answer
253 views

Reconstructing a Monthly problem: tree growth on the 2D integer lattice

I'm trying to reconstruct a problem I saw in the Monthly, years ago. Perhaps it'll look familiar to someone. In the integer lattice in the plane, we grow a tree in the following natural way: ...
58
votes
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 ...
10
votes
1answer
587 views

Results related to The Happy Ending Problem

Im giving a small talk for a combinatorics class on the Erdos-Szekeres conjecture regarding the happy ending problem (the paper is focused on recent work regarding the conjecture). I always find that ...
10
votes
1answer
437 views

A multi-dimensional Frobenius problem

Inspired by this question. Let $A$ be a subset of ${\mathbb Z}^d$ that generates the whole additive group ${\mathbb Z}^d$, and let $S$ be the additive semigroup generated by $A$. Prove that there ...
2
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5answers
206 views

Existence of lines not containing given points in general position

I remember seeing something like the following problem in the past and would like to know if it has a solution (or if I can find a source for it). Problem Given a finite set of points in the plane in ...
3
votes
2answers
398 views

Voronoi decomposition implementation in four dimensions?

I'm a software engineer and have been asked to research a Voronoi implementation in four dimensions. I'm not asking for "teh codez" but am interested in approachable tutorials on Voronoi decomposition ...
3
votes
1answer
128 views

Lower bound for a set of distances between pairs of points in a plane

There are $N>1$ points in a plane. Consider the set of all distances between pairs of the points. Let $n$ be the number of elements of this set. We know that $$n \leq {N \choose 2}.$$ What is the ...
1
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1answer
267 views

Black and White points on a grid

Some points on (integer, rectangular) grid in a plane are colored white, some black and some are not colored. In each step, one vertical or horizontal line can be selected, and all colored points on ...