# Tagged Questions

Combinatorial geometry is concerned with combinatorial properties and constructive methods of discrete geometric objects. Questions on this topic are on packing, covering, coloring, folding, symmetry, tiling, partitioning, decomposition, and illumination problems.

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### Geometric way to view the truncated braid groups?

This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question. I also asked a related question on MO, although ...
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### Size of connected regions on a randomly-colored infinite chessboard

Consider an infinite chessboard where each square is colored white with probability $p$ and black with probability $1-p$. Suppose without loss of generality that the square at $(0,0)$ is white. We ...
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### Ratio of product from one point and minimum distance

Given points $A_0,A_1,\ldots,A_n$ in the plane, let $m$ denote the minimum distance among any two points. What is the minimum value of $$\dfrac{|A_0A_1|\cdot|A_0A_2|\cdot\ldots\cdot|A_0A_n|}{m^n}?$$ ...
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### What is the shape of the convex $n$-gon which gives the maximum of a function?

Supposing that the length of every edge of the convex $n$-gon $P_1P_2$$\cdots$$P_n$ is 1, what is the shape of the $n$-gon which gives the maximum of the following function $A_n$? ...
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### Expected Number of Convex Layers and the expected size of a layer for different distributions

It is well-known that the expected number of vertices on the convex hull of random set of points in the plane distributed uniformly within a $k$-gon is $O(k\log n)$ and within a smooth shape (e.g. a ...
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### Plane tesselation, using stairs $n\times n$, is it possible?

The other day I was constructing new mathematical problems for my pupils and thought of something like this: Given the infinite sequence of "stairs" $n\times n$, constructed from $1\times1$ ...
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### Area covered by one disk more than by two disks

Given are three unit disks on the plane. Let $A$ be the area of the plane covered by exactly $1$ disk. Let $B$ be the area of the plane covered by exactly $2$ disks. Prove that $A\geq B$. ...
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### Linear Independence Game

Suppose you have a set $X$ of vectors in $\mathbb{F}_2^n$, with $|X| \ge n+1$, and consider the following game. On their turn, each player (2 player game) chooses from $X$ one vector and sets it aside ...
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### Limits to the growth of the volume of a union of spheres

Assume that $x_i$, $i=1,\ldots,m$ are points in $\mathbb{R}^n$, with the maximal distance between any two of them being at most $1$. Define $$a(r)=\mu\Bigl(\bigcup_{i=1}^m B(x_i,r)\Bigr),$$ where ...
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### Hypercube subdivision for a combinatorial problem

I have to design a combinatorial algorithm based on some simmetries of an hypercube and I'm pretty sure such a problem has already been studied. Let's start with a 3D case. Consider a cube like the ...
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### Maximum overlap of two convex bodies

Let $\Delta_n=\big\{(x_1,x_2,\ldots,x_n): \sum_i x_i \leq 1, x_i \geq 0\big\}$ denote an n-dimensional simplex. I am trying to find an $y \in \mathbb{R}^n$ such that the $y-$shifted negative simplex ...
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### Trajectories on a circular billiards table

This question is related to The case of Captain America's shield: a variation of Alhazen's Billard problem, but more focused. Let the unit disc in the plane be our billiards table, and let $C$ ...
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### Existence of fair parallel queues

I just spent a few days at a major theme park. The queue for one particular ride (involving pirates) bifurcated upon entry; the two sides wound independently through a maze and emerged next to each ...
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### How many spherical caps of height $h$ and base circle radius $a$ can cover a sphere of radius $R$?

Question How many spherical caps of height $h$ and base circle radius $a$ can cover a sphere $\mathbb S$ of radius $R \quad (R \gg a)$? What I have thought so far Since the area of the ...
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### Maximum number of acute triangles

Given $n$ points on the plane, no three of which are collinear, what is the maximum number of acute triangles formed by them? [Source: Based on Hungarian competition problem]
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### Given two lists of similar orthogonal matrices with common “conjugator”, determine that conjugator

Here's a question related to a long-time personal research project in combinatorial geometry. Suppose I have two lists of similar $n$-by-$n$ orthogonal matrices $P_i$ and $Q_i$, and suppose I know ...
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### Realisations of associahedra

I seem to have lost the reference to a realisation I am interested in. Hopefully someone can steer me to a paper that fully explains the realisation. For the case $K_2$(the 5-gon) the following ...
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### Fitting cubes inside a bigger cube

Suppose the sum of the volumes of $n$ cubes is 1. Then no matter what $n$ is I need to prove they can be put inside a cube of volume $\leq 2$ such that they do not overlap. I am totally going nuts ...
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### Internal angle of a vertex of degree $d$ in $\mathbb{E}^2$ and $\mathbb{S}^2$

I am currently working on determining the maximum number of times the minimum spherical distance can occur among $n$ points in $\mathbb{S}^2$, and I have the following question. In $\mathbb{E}^2$, ...
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### Combinatorics problem involving n-dimensional space

Consider a set of more than $\frac {2^{n+1}} {n}$ points $(n>2)$, chosen from the $2^n$ points of the $n$-dimensional space which have the coordinates $\{ \pm1, \pm1, ..., \pm1 \}$. Show that ...
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### Number of simplices of convex hull of points on a $d$-sphere.

I was discussing this with my professor the other day and he left me to figure out. And I can't for the life of me, figure out why this is so. I would appreciate what I should look into rather than an ...
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### A circle can include all but one of n points, but which one can it be?

The answers to the question "Circle enclosing all but one of n points" demonstrate that, given $n$ points, it is possible to construct a circle such that all but one of the points is inside the circle ...
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### Triangulations of combinatorially equivalent polytopes

I am wondering which relation(s) there are between triangulations of combinatorially equivalent polytopes that use no new points: Let $P,Q$ be a $n$-polytopes such that their face lattices are ...
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### Hexagons share interior points

Can we draw infinitely many hexagons, not necessarily convex, on the plane so that any three of them share a common interior point, but no four of them does? For four hexagons this is possible, using ...
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### What is the optimal tiling of a regular n-gon in the plane?

I want to tile the plane with equal-sized regular polygons of $n$ sides. Obviously for some $n$, the tiles will be able to tessellate and cover the whole plane (e.g triangles, squares, hexagons) I ...
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### Points on the circle

We have $n$ points on the unit circle. What is the best configuration if we want to maximize the sum of the pairwise distances of the given points?
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### Number of connected sets intersecting a given set in $\mathbb{Z}^d$

Let $A \subset \mathbb{Z}^d$ and let $|A|$ be its cardinality. Let $F_n(A)$ be the number of connected sets of $\mathbb{Z}^d$ having cardinality $n$ and intersecting $A$ in at least one site. Assume ...
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### Center of mass of vertices without enumeration?

Given a $n$-dimensional convex polytope defined by $A x\leq b$ and $A_{eq} x = b_{eq}$, is there an efficient way to determine the average coordinates of all vertices without enumerating them? (As if ...
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### Maximum number of edges in a subgraph of hypercube

Let $H_n$ is an $n$-dimensional hypercube, $|V(H_n)|=2^n, |E(H_n)|=n2^{n-1}$. Let $M\subset V(H_n), |M|=2^k, 1\le k<n$, and $G_M$ is a subgraph of $H_n$ induced by $M$, $V(G_M)=2^k$. Prove that ...
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### An art gallery problem

An art gallery has the shape of a simple $n$-gon. Find the minimum number of watchmen needed to survey the building, no matter how complicated its shape be. I failed to solve the problem. Please help ...
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### Fill a rectangle with squares

How many ways are there to fill a $m\times n$ rectangles with squares that have integer side lengths. Both $m$ and $n$ are integers.
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### Red and blue balls lined up

On a plane, is it possible to arrange $6$ red points and $6$ blue points such that No $2$ points coincide. For any line containing two or more points, not all the points on that line are of the same ...
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### Neighbor-full partition of $\{0,1\}^n$

What is the partition of $\{0,1\}^n$ with each set connected and neighboring each other that has the maximum number of elements? (which we call $k(n)$) We say $A$ and $B$ are neighbors if their ...
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### Is it possible to choose 10 points from 20000 with that property?

A set of $20000$ points is chosen in a ball of radius $6$ in $\mathbb{R}^3$. Do there exist $10$ ones of these points s. t. all the distances between them are less than or equal to $1$?
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### Why is unit circle not sufficient to bound the partial sums?

I want to find vectors $\textbf{v}_1, \dots,\textbf{v}_n$ in $\mathbb{R}^2$ with that $\sum_{i=1}^n\textbf{v}_i=\textbf{0}$ and $\Vert \textbf{v}_i\Vert\leq 1$ for all $i=1,\dots,n$, such that for ...
This question is from a Japanese contest: Let $S$ be a set of 2002 points in the coordinate plane, no two of which have the same $x$- or $y$- coordinate. For any two points $P,Q$ in $S$ consider ...