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12
votes
2answers
216 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
2answers
236 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
8answers
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
1answer
462 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
7answers
757 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
0answers
134 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
3answers
647 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
2answers
240 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
0answers
136 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
4answers
468 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 ...
4
votes
1answer
318 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
3answers
605 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
0answers
225 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
3answers
688 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
1answer
235 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
0answers
402 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 ...
6
votes
2answers
499 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
1answer
325 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
2answers
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 ...
19
votes
1answer
443 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
1answer
249 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
2answers
310 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 ...
1
vote
1answer
273 views

Large subgroups of the Hamming cube

Let's consider the abelian group $\mathbb{Z}^N_2$ equipped with the Hamming metric (the hypercube). Suppose I have a subgroup of this hypercube (not necessarily a subcube) which is generated by a set ...
2
votes
2answers
311 views

Determining the vector space for application of Cauchy Schwarz

In the paper "On the Erdős distinct distance problem in the plane" by Larry Guth and Nets Hawk Katz, http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.4105v1.pdf they define the functions $d(P)$, ...
2
votes
0answers
245 views

Literature Request: Genus Two Partition Functions

Apologies in advance for what will surely sound like a, "well why can't you just Google it" question, but I'm struggling to find good literature that presents the basic construction of genus two ...
1
vote
1answer
188 views

Equivariant homology of Hilb and torus stable curves

The torus equivariant homology of a compact homogeneous variety $G/B$ may be described by using the "skeleton" consisting of torus fixed points and the torus invariant curves that connect them. ...
7
votes
0answers
292 views

A combinatorial approximation functor sSet->qCat

Let $sSet_J$ denote the category of simplicial sets equipped with the Joyal model structure. Simply by the fact that $sSet_J$ is locally presentable and its class of anodynes ($\neq \mathbf{Cof} \cap ...
6
votes
0answers
581 views

An Ex functor for the contravariant homotopy structure

I'm going to slack on the background and get to the point: Is there a good notion of an $Sd/Ex$ adjunction for $sSet/S$ equipped with the contravariant model structure (cofibrations are monomorphisms ...
22
votes
2answers
914 views

A geometric Ramsey problem

The following problem seems like one to which the answer could well be known: if so, I'd be interested to have a reference. How large does n have to be such that among any n points in the plane you ...
6
votes
0answers
420 views

Historical and terminological questions about Dan Kan's Ex functor and its relation to the classical case of simplicial complexes

Recall that we may define a functor $\xi:\Delta\to \operatorname{Poset}$ sending a simplex $[n]$ to the set of monotone injections $[k]\hookrightarrow [n]$ for $k\geq 0$ (effectively, $k\leq n$ as ...
53
votes
5answers
2k views

Does every polyomino tile R^n for some n?

This is a question posed by Adam Chalcraft. I am posting it here because I think it deserves wider circulation, and because maybe someone already knows the answer. A polyomino is usually defined to ...
52
votes
1answer
2k views

How many cubes cover a bigger cube?

How many $n$-dimensional unit cubes are needed to cover a cube with side lengths $1+\epsilon$ for some $\epsilon>0$? For n=1, the answer is obviously two. For n=2, the drawing below ...
2
votes
1answer
303 views

When are nontopological bistellar flips manifold-preserving?

A topological bistellar flip is the term used by Dougherty, Faber, and Murphy to describe a bistellar flip that does not cause any face of a complex to be duplicated. Suppose we consider a ...
2
votes
1answer
502 views

$n$ lines in general position; there are $n-2$ small triangles

Suppose we have $n$ lines in general position in the plane. Prove that there are at least $n-2$ ''small'' triangles. Here a "small" triangle is a triangle that is not contained in any larger triangle. ...
8
votes
2answers
642 views

Are Penrose tilings universal? Do aperiodic universal tilings exist?

Consider a tiling of the plane using tiles of at least two types (e.g, a Penrose tiling such as that shown at the bottom of this question, which tiles the plane with two types of tiles). List the tile ...
2
votes
2answers
508 views

Counting integral points of a polytope in R^3 (the c_1 coefficient of Ehrhart polynomial)

(Sorry I'm outsider in this field.) I need to count the number of integral points in a convex polytope in $\mathbf{R}^3$. The cones in the dual fan are not necessarily regular (does it create any ...
13
votes
0answers
1k views

Minimum tiling of a rectangle by squares

Given the $n\times m$ rectangle, I want to compute the minimum number of integer-sided squares needed to tile it (possibly of different sizes). Is there an efficient way to calculate this?
14
votes
3answers
1k views

Representation of vectors in $\mathbb{R}^2$ via differences of small vectors.

Is the following fact true? Let $v_1,\ldots, v_k \in \mathbb{R}^2$, $\|v_i\|\leq 1$, be vectors that add up to zero. Does there exist a permutation $\sigma\in S_k$ and vectors $w_1,\ldots, w_k ...
6
votes
2answers
455 views

Presentation of the cohomology of generalized flag varieties as graded ranks of rings of symmetric polynomials

Hello! Let $n,m\geq 0$ be integers. If I understand it correctly, there is the following description of the cohomology of the complex Grassmannian $\text{Gr}(m+n;m)$: denote by $\text{Sym}(n,m)$ the ...
2
votes
1answer
179 views

Constructing affine hypersurfaces with one singularity

This is a followup to my previous poorly-worded question. Consider a finite collection of points $S \subset \mathbb N^n$ lying in a hyperplane $H$. These points define exponents of a collection of ...
4
votes
1answer
364 views

Secondary fans and Stanley Reisner ideals

Consider a collection of points $S \subset \mathbb R^d$. I would like to understand all possible fans $\Sigma$ whose support is the cone over $S$: $|\Sigma| = cone(S)$. I have heard that the ...
1
vote
0answers
232 views

Any 2-coloring of the scope of equilateral triangle must create a monochromatic rectangular triangle ?

Hello All, I'm trying to prove something that we know that is correct but can't find how to do it or any example to show how to solve it. Any 2-coloring of the scope of equilateral triangle must ...
5
votes
2answers
327 views

Up to projectivities, which configurations of four lines in $\mathbb{P}^3$ can one distinguish?

Background I am interested in the projective classification of reduced curves of degree four in $\mathbb{P}^3(\mathbb{R})$ (and more generally of degree $n+1$ in $\mathbb{P}^n(\mathbb{R})$). More ...
3
votes
1answer
198 views

Can any vertex remain when removing halfspaces from a projectively transformed polytope?

Let P be a simple polytope defined as an intersection of n halfspaces. A facet F of P, supported by halfspace H, is removable if the intersection of the remaining (n-1) halfspaces is bounded. F is ...
2
votes
1answer
368 views

When can a folded polygon be isometrically (locally) embedded into R^3 ?

I am interested in 3-D representations of various things that naturally live in a non-simply-connected compact surface. There is the usual way of producing a compact surface of any orientable or ...
22
votes
3answers
2k views

Covering a circle with red and blue arcs

We have a circle and two families of $n$ red arcs and $n$ blue arcs, positioned on the circle so that every two arcs of different colors intersect. Can one show that there is a point in the perimeter ...
10
votes
1answer
378 views

Random geometric graphs and spanners

I would grateful to learn of work mixing random geometric graphs with random graphs under the Erdős-Renyi model, and in particular concerning spanners. Select $n$ points uniformly at random from the ...
4
votes
4answers
514 views

Upper bound for the number of subsets of N points which exhaust their convex hull

Hello. Looking at a set S of N points in the plane, I call a subset B of S "legal" if the set of points contained in the convex hull of B is exactly B itself. In other words, a subset B of S in legal ...
11
votes
3answers
1k views

When are Ehrhart functions of compact convex sets polynomials?

Given a lattice $L$ and a subset $P\subset \mathbb R^d$, we define for each positive integer $t$ $$f_P(L,t)=|(tP\cap L)|$$ the number of lattice points in $tP$. Let's say $P$ is nice if $f_P(L,t)$ is ...
8
votes
3answers
865 views

Which finite groups are generated by n involutions?

One of the interesting problems in abstract polytope theory is to determine, for a given finite group, when that group is the automorphism group of a regular abstract polytope. This is equivalent to ...