The combinatorial-geometry tag has no wiki summary.

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votes

**2**answers

183 views

### Generators of a 2D lattice

Dear MO_World,
I'm hoping someone can point me towards a reference for something. I have an invertible $2\times 2$ matrix, $A$, with real entries such that for both of the rows, the entries are ...

**10**

votes

**3**answers

619 views

### Combinatorial analogues of curvature

There appear to be many "combinatorial" definitions of curvature as applied to finite simplicial (or regular CW) complexes. For instance, we have the ideas of Cheeger, Muller and Schrader, of Forman ...

**13**

votes

**1**answer

483 views

### Partitioning the vertices of an n-cube with random hyperplane cuts

An evolutionary biologist asked me a question which boils down, at least in part, to what seems to me an interesting question of combinatorial/probabilistic geometry.
It is an old chestnut of a ...

**3**

votes

**1**answer

257 views

### Grading a non-graded poset as squeezed as possible

Here is a curiosity question (motivated by the recent revamp of ranked-poset routines in Sage).
Let $P$ be a finite poset. We look for a family $\left(a_p\right)_{p\in P}$ of real numbers summing up ...

**2**

votes

**0**answers

129 views

### A relation on triplets of points in the plane

This question is a follow up of my previous one (Planar sets closed under intersection of circles, Planar sets closed under intersection of circles) and is motivated by G. Zaimi's answer ...

**1**

vote

**1**answer

108 views

### Generalizing the internal angle of a graph in $\mathbb{E}^2$ to $\mathbb{S}^2$

I am currently working on research involving packing problems and am finding myself needing the tools from Combinatorial Geometry (in particular, I've been reading Pach and Agarwal's book on the ...

**4**

votes

**1**answer

287 views

### Planar sets closed under intersection of circles

Let $P$ be the plane with a point at infinity. By plane, I mean the Euclidian plane, and therefore it has circles. A line is also a circle, though its center is at infinity. If $A\subset P$ has ...

**45**

votes

**2**answers

2k views

### vector balancing problem

I believe the solution posted to the arXiv on June 17 by Marcus, Spielman, and Srivastava is correct.
This problem may be hard, so I don't expect an off-the-cuff solution. But can anyone suggest ...

**3**

votes

**1**answer

316 views

### from affine matroid to measures

Let $S$ be an arbitrary finite spanning subset of $\mathbb{R}^d$ of cardinality $N$. Let
$W(S)$ be the formal $\mathbb{R}$-vector space generated by all $d$-dimensional
simplices (i.e. bases of the ...

**0**

votes

**1**answer

339 views

### On the number of lines of given points

Hi all, I have a question Concerning Beck's theorem. I have read it from http://en.wikipedia.org/wiki/Beck%27s_theorem and I have two questions :
I suppose Beck's theorem doesn't hold when instead ...

**9**

votes

**2**answers

290 views

### The area of the intersection of convex sets with prescribed pairwise intersections

Consider two numbers $a>b>0$. Let $A_1,A_2,A_3$ be three convex sets in ${\mathbb R}^2$ such that $\mu(A_i)=a$, $\mu(A_i\cap A_j)=b$ ($\mu$ is the usual measure on ${\mathbb R}^2$). What is the ...

**15**

votes

**3**answers

986 views

### Is Euler characteristic of a simplicial complex upper bounded by a polynomial in the number of its facets ?

What is the best upper bound known on the (absolute value of) the
Euler characteristic of a simplicial complex
in terms of the number of its facets ?
In particular, I am interested in proving or ...

**4**

votes

**2**answers

477 views

### On the joints problem in finite fields

The original version of the so-called "joints problem" consists of the following:
Let $L$ be a set of lines in $\mathbb{R}^{3}$. Determine the maximum number of "joints" determined by these lines, ...

**26**

votes

**2**answers

614 views

### Careless packing

The sequence $\frac{1}{2}, \frac{1}{2\cdot 2}, \frac{1}{3\cdot 4}, \frac{1}{4\cdot 6}, \frac{1}{5\cdot 8}, \frac{1}{6\cdot 10},\ldots$ has a curious property, as follows:
a) the series with these ...

**2**

votes

**0**answers

235 views

### Erdős-Szekeres empty pseudoconvex $k$-gons

I am wondering if the
Erdős-Szekeres
empty convex $k$-gon question has a different answer if
convexity is replaced by a pseudoline-version of convexity.
The empty convex $k$-gon question
is a variant ...

**26**

votes

**2**answers

868 views

### Bodies of constant width?

In two-dimensional case one can generalize figures of constant width as figures which can rotate in a covex polygon.
Here is one example which can be used to drill triangular holes:
I would like to ...

**6**

votes

**2**answers

213 views

### convex polytopes with many faces and edges but few cells and vertices

For a convex polytope $P$ in $\mathbb R^4$, denote by $N_0,N_1,N_2,N_3$ respectively the number of vertices, edges, faces, cells. By Euler's formula, we know $N_0+N_2=N_1+N_3$, which means there is a ...

**7**

votes

**1**answer

204 views

### Integer points in dilations of a disk of volume one

The following question is related to
When are Ehrhart polynomials polynomials?. For a positive integer $n$,
let $i(D,n)$ be the number of
integer points in the disk $x^2+y^2\leq n^2/\pi$, so $i(D,n) ...

**13**

votes

**2**answers

2k views

### How many vertices/edges/faces at most for a convex polyhedron that tiles space?

I wonder if this problem has already been examined before:
Consider a convex polyhedron that tiles $\mathbb R^3$. What is the maximum of vertices/edges/faces that such a polyhedron can have?
...

**1**

vote

**1**answer

383 views

### Combinatorial optimization and graph coloring

I am considering the following problem:
(i) Fix $n$ and color the edges of $K_n$ red and blue arbitrarily.
(ii) Let $M$ be the set of monochromatic triangles in $K_n$ and define $g:M\rightarrow ...

**0**

votes

**0**answers

100 views

### Question regarding contiguous forms

I read about contiguous forms in Achill Scürmann's thesis on positive quadratic forms. I am wondering about one aspect of the Voronoi algorithm presented in there, that enumerates all arithmetically ...

**1**

vote

**0**answers

137 views

### Traversing perfect quadratic forms

I am wondering whether there is an efficient algorithm to traverse all the $N\times N$ perfect quadratic forms $Q$ inside the polyhedron $e_j^T Q e_j \geq 1$, $j = 1\ldots m$, where $e_j$ are some ...

**4**

votes

**0**answers

139 views

### Slices of Simplices that are Simplices, Reference?

I am trying to find a reference for the following fact. It is elementary and not hard to prove, but I haven't been able to find the question treated anywhere.
Let $A$ be an $l\times n$ matrix with ...

**2**

votes

**2**answers

362 views

### the minimal diameter of a quadrilateral

Let a convex quadrilateral ABCD with perimeter 1，d is the maximum of AB,AC,AD,BC,BD,CD,prove that d is not less than 1/3
we can prove that parallelogram ABCD with perimeter 1，than one of AC,BD is ...

**1**

vote

**3**answers

284 views

### Enumerating Perfect Lattices

I have a question regarding enumeration of perfect lattices/quadratic forms. In the thesis of Achill Schürmann http://fma2.math.uni-magdeburg.de/~achill/public/habil.pdf there is an algorithm called ...

**4**

votes

**1**answer

558 views

### Why is the Street nerve of the Gray tensor product $[1]\otimes [1]$ isomorphic to $[1]\times [1]$

Recall that given two strict ω-categories $A$ and $B$, their lax Gray tensor product $A\otimes B$ is sent to the Verity-Gray tensor product of their associated complicial sets ...

**23**

votes

**3**answers

653 views

### Largest possible volume of the convex hull of a curve of unit length

What is the largest possible volume of the convex hull of an open/closed curve of unit length in $\mathbb{R}^3$?

**7**

votes

**1**answer

345 views

### Building a polyhedron from areas of its faces

Is there a known algorithm which, given a finite multiset (unordered list) of integers $A$, returns a yes/no answer for ...

**12**

votes

**2**answers

222 views

### Smallest containing simplex

Let $V_n$ be the least real number such that for every convex subset of $\mathbb{R}^n$ with hypervolume $1$ there is a containing simplex with hypervolume $V_n$.
What is known about $V_n$? Is there a ...

**7**

votes

**2**answers

237 views

### $\kappa$-coloring of $\mathbb{R}^2$ and triangle with area 1

What is the largest cardinal number integer $\kappa$ such that every $\kappa$-coloring of $\mathbb{R}^2$ contains a triangle with area 1 and all vertices of the same color?

**19**

votes

**8**answers

1k views

### Higher-dimensional Catalan numbers?

One could imagine defining various notions of higher-dimensional Catalan numbers,
by generalizing objects they count.
For example, because the Catalan numbers count the triangulations of convex ...

**1**

vote

**1**answer

473 views

### On lattice points “far inside” convex lattice polygons

Let $\mathcal{P}$ be a convex lattice polygon with $n$ vertices and let $\mathcal{L}$ be the set of all lattice points inside $\mathcal{P}$. For every $n \geq 5$, does there exist a point in ...

**8**

votes

**7**answers

766 views

### Omniscient bots gathering on $\mathbb{Z}^2$

There are $N=n^2$ "bots" on distinct integer lattice points in the plane.
Each knows the positions $p_i$ of all bots, and each has unlimited (private) memory.
Each executes the same algorithm ...

**1**

vote

**0**answers

142 views

### When is the conical hull of a finite set of vectors a subset of the space? (and tilings)

Consider a hypercube in n-dimensions, and take some projection down to an m-dimensional subspace. Now take all vertices and m-1 dimensional facets visible from some direction outside the projection. ...

**9**

votes

**3**answers

718 views

### Point sets in Euclidean space with a small number of distinct distances

It is well known and not hard to prove that the regular simplex in n-dimensions is the only way to place n+1 points so that the distance between distinct pairs of points is always the same. My general ...

**4**

votes

**2**answers

251 views

### Isostatic graphs and the Henneberg conjecture

I have been reading "Combinatorial Rigidity" by Graver, Servatius and Servatius and I am interested in their chapter on rigidity in dimension $\geq$ 3. I have two questions.
What is the current ...

**2**

votes

**0**answers

138 views

### Minimally 6-connected 3D discrete lines that are convex lattice sets

There are several definitions of 3D discrete lines, e.g. http://diwww.epfl.ch/w3lsp/publications/discretegeo/nratddl.html , http://dx.doi.org/10.1007/978-3-642-19867-0_4 . However, I know of none that ...

**1**

vote

**4**answers

483 views

### Decomposition of inner horns

Hi guys, I have a question. Prove or disproof the statement:
Any inner horn $\Lambda[n,k],0< k< n $ admits a filtration $\mathbf{n}<\cdots<\Lambda[n,k]$, such that each step is filling an ...

**5**

votes

**1**answer

327 views

### Convex tilings of the plane

For convex polyhedra you have Steinitz's theorem characterizing them as the 3-connected planar graphs. My question is not about spheric tilings, but about periodic tilings of the euclidean plane. Is ...

**7**

votes

**3**answers

631 views

### Not quite regular polyhedra

Take a naive interpretation of regular polyhedra:
All vertices (including epsilon ball) congruent
All edges congruent
All faces congruent
We can now find interesting families by removing one ...

**3**

votes

**0**answers

229 views

### Polygon illumination with perturbed reflections

Here is a variation on the classical polygon illumination problem. For $c \geq 0$ we say that a mirror has reflection index $c$ if whenever a ray hits the mirror with angle of incidence $\alpha$ then ...

**9**

votes

**3**answers

725 views

### Polytopes with few vertices.

Suppose I have a convex polytope in $\mathbb{R}^d$ which I know has few vertices (in the case which prompted this question, I seem to have a polytope in $\mathbb{R}^9$ which has sixteen vertices). Is ...

**6**

votes

**1**answer

238 views

### Simplices in convex polytopes

This question is a direct generalization of:
Counting the (additive) decompositions of a quadratic, symmetric, empty-diagonal and constant-line matrix into permutation matrices
Given a convex ...

**4**

votes

**0**answers

412 views

### Intersection of pencils in $\mathcal{R}^2$

Consider $9n$ pencils through non-collinear points $p_1, \ldots , p_{9n}$ in $R^2$ each consisting of at most $n$ concurrent lines. Define the intersection $S$ of these pencils to be the set of points ...

**7**

votes

**2**answers

517 views

### Maximal number of edges and triangular cells for n points in a triangular lattice

Consider a subset of $n$ points in an equilateral triangular lattice. Draw all the edges between nearest-neighbor points.
What is the maximum, over all such subsets, of the number of edges? This ...

**4**

votes

**1**answer

335 views

### A “join” of ω-categorical simplices

Recall that the category of level trees $\mathcal{T}$ is defined to be the category $[\mathbb{N}^{op},\Delta_a]$, where $\Delta_a$ is the skeleton of the category of finite possibly empty linearly ...

**10**

votes

**2**answers

1k views

### Subdivision of triangles into congruent triangles

Recently some old notes of mine have gotten me to thinking about the problem of subdividing a triangle into $N$ smaller triangles, all congruent to one another. A little thought shows the following ...

**20**

votes

**1**answer

469 views

### Voronoi cell of lattices with the same profile

Definition 1. Given a body $V$ in $\mathbb R^n$,
the function $p_V\colon \mathbb R_+\to \mathbb R_+$
$$p_V(r)=\mathop{\rm vol} [V\cap B_r(0)]$$
will be called profile of $V$.
Definition 2. Define ...

**0**

votes

**1**answer

256 views

### [Matrices over Z] - An algorithm for calculating the diagonal with elementary operations

Dear mathoverflow,
Let
$
\left(
\begin{array}{cc}
a & b \newline
c & d
\end{array}
\right)
$
be a matrix with $a, b, c, d \in \mathbb{Z}$, $\gcd(a,b,c,d) = 1$ and $ad - bc = \pm N$, with $N ...

**4**

votes

**2**answers

312 views

### Approximate search space on a 5x5x5 cube with 3 different possible classes?

Hey all,
I read the meta, and I realize this question might be pretty elementary for this site, but I'm having trouble computing this, and I know it won't take too much insight for someone to give me ...