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6
votes
1answer
312 views

Groups and pregeometries

Definition. For an infinite structure $\mathcal{A}$ and $cl : P(dom(\mathcal{A})) \longrightarrow P(dom(\mathcal{A}))$ , we say that $(\mathcal{A}, cl)$ is a structure carrying an $\omega$-homogeneous ...
0
votes
0answers
90 views

Graphics for the Whitehouse simplicial complexes ( related to inversion and moduli spaces)?

Are there some decent graphics for the Whitehouse simplicial complex of sixth order, or some type of dual, or other rep (positive Bergman complexes of complete graphs (?)) that would help to visualize ...
23
votes
1answer
331 views

what-if.xkcd.com: stabbing (simply connected) regions on the 2-sphere with few geodesics

In the latest what-if Randall Munroe ask for the smallest number of geodesics that intersect all regions of a map. The following shows that five paths of satellites suffice to cover the 50 states of ...
22
votes
2answers
505 views

Random points on the unit sphere

Suppose you have $n$ points picked uniformly at random on the surface of $\mathbb{S}^d,$ and let the volume of the convex hull of these points be $V_{n, d}.$ Clearly, $V_{n, d}$ converges to the ...
6
votes
2answers
170 views

Pictures of the von Neumann polytope

Are there any graphic portrayals of von Neumann polytopes in low dimensions?
2
votes
1answer
104 views

Distance from constant width bodies

EDIT As @David has observed, my conjecture was clearly wrong for $\ n:=2.\ $ Let me still give it a chance for $\ n\ge 3$. I'll call a family $\ F\ $ of bound closed convex subsets of $\ \mathbb ...
2
votes
2answers
101 views

Maximum possible number of similar three-colored triangles

I want to maximize the number of similar triangles with vertices from three fixed sets, one vertex from each set. For example, if you fix two points $X$, $Y$ (i.e. two sets with only one member), then ...
12
votes
0answers
726 views

Hrushovski's Construction

Zilber expressed a conjecture for $\aleph_{1}$- categorical theories (In the 80s). Zilber's Conjecture: The geometry of any $\aleph_{1}$- categorical structure is one of the following: (a) Trivial ...
19
votes
1answer
605 views

Monomer-Dimer tatami tilings need better relationships with other math. Summary of results.

A monomer-dimer tiling of a rectangular grid with $r$ rows and $c$ columns satisfies the \emph{tatami} condition if no four tiles meet at any point. (or you can think of it as the removal of a ...
22
votes
3answers
649 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 $\mathbb{R}^3$?
8
votes
1answer
224 views

Largest convex hull of a unit length path

What is the largest area possible for the convex hull of a path of unit length lying on a plane? For what paths is that largest area attained?
0
votes
0answers
110 views

“Open Points” in the 1983 proof of Szemerédi-Trotter theorem

I was reading through the 1983 paper "Extremal Problems in Discrete Geometry" and I was confused about the definition of "open point" appearing in this paper. By this point in the paper, the authors ...
6
votes
1answer
294 views

Triangle (constrained number, rather than shape) packing?

Are there any interesting results on optimal packings in the plane using a fixed number of triangles (without a fixed size or shape constraint)? For instance, what's the maximum area packing of the ...
1
vote
1answer
140 views

Is there a “Bipartite” Szemeredi-Trotter theorem?

One version of the Szemeredi-Trotter theorem states the following: Given a set of $L$ lines in the plane, the number of points incident to at least $k$ lines is bounded above by a constant times $L/k ...
7
votes
1answer
88 views

Dropping altitudes to achieve nonobtuse planar triangulations: finite or infinite?

Given a planar triangulation of (say) a convex region, imagine the following process to convert it to a triangulation with no obtuse angles: Pick an arbitrary obtuse angle at vertex $a$ of ...
5
votes
2answers
226 views

Put P inside Q! polygons/polyhedra

We have two Polygons/Polyhedra P and Q. Does there exist a polynomial time algorithm to decide if we can put P (using translation and rotation) inside Q or not? First think about the case of which P ...
5
votes
1answer
183 views

Small remarkable matroids

Working on phased matroids (a generalization of oriented matroid to the complex case) I've found an interesting formula for computing the inner Tutte group (and, hence, all the Tutte groups) ...
10
votes
2answers
457 views

Cut and Fold Polyhedron!

I have two convex polyhedra such that their sums of side areas are equal. It is true that I can cut one of them and flatten it on the plane, then fold the flattened polygon to reach the other ...
2
votes
1answer
126 views

intersection of the unit cube and a hyperplane containing the main diagonal

Let $A$ be a linear $m$-dimensional subspace of $\mathbf{R}^n$ $m < n$, containing the point $(1,1,\ldots,1) \in \mathbf{R}^n$, and consider the intersection of $A$ and the unit cube $\Delta_n$ ...
1
vote
0answers
32 views

Generalizing Concepts of Planar Euclidean Geometry to Symmetric TSP-Instances

To me it seems possible, to successfully look at symmetric TSP instances from a geometry-point of view. Examples are: the diagonals of the convex hull of a set of points in the euclidean plane; ...
3
votes
1answer
101 views

Polygonal Venn diagrams

Suppose that the interiors of $n$ $m$-sided planar simple closed polygons generate a $\sigma$-algebra $A$. How many atoms can $A$ possess, at the most? Failing an exact answer, how about good ...
0
votes
0answers
63 views

Toric morphism fiber and kernel dimensions

Given a morphism between two smooth toric varieties $f: X \rightarrow Y$, is the dimension of the kernel of $\mathrm{d}f$ at any point $p \in X$ equal to the dimension of the fiber at $f(p) \in Y$? ...
11
votes
2answers
291 views

When does a set of collinearity conditions imply collinearity of all of the points?

Suppose we have a set of $n$ points $\{X_1,X_2,\dots,X_n\}$ in the real plane and $\mathcal{A}$ a family of subsets of $\{1,\dots,n\}$. By a "set of collinearity conditions for $\mathcal{A}$" we mean ...
1
vote
1answer
154 views

Positroids and Totally Nonnegative Complex Grassmanian

Recently I begin working on matroids, in particular to a generalization of oriented matroids to the complex case. I found on arxiv the following interesting articles: 1)Alexander Postnikov: Total ...
19
votes
3answers
763 views

Visibility of vertices in polyhedra

Suppose $P$ is a closed polyhedron in space (i.e. a union of polygons which is homeomorphic to $S^2$) and $X$ is an interior point of $P$. Is it true that $X$ can see at least one vertex of $P$? More ...
4
votes
1answer
177 views

Light inside a polyhedron

I have two questions the same as Mostafa's Question: Visibility of vertices in polyhedra Suppose $P$ is a closed polyhedron in space (i.e. a union of polygons which is homeomorphic to $S^2$) and $X$ ...
15
votes
4answers
2k views

Why is complex projective space triangulable?

In an exercise in his algebraic topology book, Munkres asserts that $\mathbf{C}P^n$ is triangulable (i.e., there is a simplicial complex $K$ and a homeomorphism $|K| \rightarrow \mathbf{C}P^n$). Can ...
12
votes
3answers
511 views

Is {6,3,7} an 'ultrahyperbolic' Coxeter group?

These pictures, drawn by Roice Nelson, are attempts to visualize a geometry having as symmetries the {6,3,7} Coxeter group, by which I mean the one coming from the Coxeter diagram ...
24
votes
3answers
1k views

Can one recover the smooth Gauss Bonnet theorem from the combinatorial Gauss Bonnet theorem as an appropriate limit?

First let me state two known theorems. Theorem 1 (for smooth manifolds): Let $(M,g)$ be a smooth compact two dimensional Riemannian manifold. Then $$ \int \frac{K}{2 \pi} dA = \chi (M) $$ where $K$ ...
6
votes
2answers
115 views

Uniqueness of an equilateral triangle decomposition into three similar polygons, exactly two congruent

A favorite fun problem I give to students and (even non-mathematical) friends is the following: Find a decomposition of an equilateral triangle into three similar polygons, exactly two of which are ...
1
vote
0answers
84 views

What is the projective dual of a planar graph?

Everybody learns the usual definition of the dual of a planar graph when edges are preserved and faces are mapped to vertices. Everybody learns the projective duality. What if we apply it to a ...
2
votes
1answer
92 views

Relation between non-integral polytopes, integrally closed polytopes and polynomial Erhart quasi-polynomials

By lattice points, I will always mean points in $\mathbb{Z}^n$ and all polytopes here are convex rational polytopes. If $P$ is an integral polytope, the counting function for the number of lattice ...
0
votes
0answers
106 views

Generalized Sphere Kissing Problem

I am currently researching discrete geometry and I am in need of an upper bound on a generalized kissing number in 3-dimensions dependent upon a parameter $\eta$ which is the radii of spheres touching ...
4
votes
5answers
428 views

Lattice points in dilated polytopes and sumsets

Let $P$ be an integral polytope, that is, the convex hull of some points in $\mathbb{N}^d$. Let $p_1,\dots,p_m$ be all lattice points in $P$. Question: What is the condition on $P$ that guarantees ...
27
votes
3answers
2k views

Is the ratio Perimeter/Area for a finite union of unit squares at most 4?

Update: As I have just learned, this is called Keleti's perimeter area conjecture. Prove that if H is the union of a finite number of unit squares in the plane, then the ratio of the perimeter and ...
9
votes
1answer
273 views

Maximum number of Vertices of Hypercube covered by Ball of radius R

Let $R>0$ be given and let $H^n$ be the unit hypercube in $\mathbb{R}^n$. The problem I am facing is to find the maximum number of vertices of $H^n$ which can be covered by a closed $n$-dimensional ...
6
votes
1answer
107 views

How “accidental” are equalities between parts of Ehrhart quasi-polynomials? When do they persist to Euler-Maclaurin?

Background What I think of Ehrhart theory (http://en.wikipedia.org/wiki/Ehrhart_polynomial) asserts that if we take a lattice polytope $P$, and count the number of lattice points in the $t$th ...
20
votes
1answer
468 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 ...
1
vote
0answers
68 views

Separating unit disks by circles

This is inspired by the recent question about separating unit disks by lines, which I will refer to as the "line case". Replacing "line" by "circle" adds one degree of freedom, and I'm wondering if ...
32
votes
0answers
473 views

Extending a line-arrangement so that the bounded components of its complement are triangles

Given a finite collection of lines $L_1,\dots,L_m$ in ${\bf{R}}^2$, let $R_1,\dots,R_n$ be the connected components of ${\bf{R}}^2 \setminus (L_1 \cup \dots \cup L_m)$, and say that {$L_1,\dots,L_m$} ...
7
votes
1answer
227 views

Separating unit disks by lines

Given $n\ge 2$. For a real $d>2$, consider a constellation $C$ of $2n$ disks of radius $1$ in the plane such that $h(C)$, the minimal distance between any two of their centers, is equal to $d$. Let ...
1
vote
0answers
90 views

On 'Very Movable' Geometric Configurations (Configurations with a large degree of freedom)

Let $C$ be an $(n_r, b_k)$ combinatorial configuration that admits a geometric realization in the plane. I'm interested in the maximum number of points/lines $M$ of $C$ we can place in general ...
3
votes
0answers
402 views

How many combinations does Android pattern have? [closed]

Rules- 1) At-least 4 and at-max 9 dots must be connected. 2) There can be no jumps 3) Once a dot is crossed, you can jump over it.
2
votes
0answers
36 views

How many regions are created by the set of all hyperplanes defined by a set of points?

If we have a set of points X in d-dimensional euclidean space, and we look at the set of all hyperplanes that are defined by any subset Y of X (in the sense of being the unique hyperplane containing ...
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 ...
1
vote
0answers
102 views

intersection points and lines, part two

Two years ago I asked a question about finding couples of integers (p,l) such that one may draw l lines on the euclidean plane, creating exactly p intersection points, and I got very interesting ...
4
votes
2answers
182 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 ...
5
votes
1answer
222 views

Regular lattice polygons

Suppose I want to construct an $N$-gon in the plane whose vertices are integer lattice points, and which is close to a regular $N$-gon (which means, the ratio of longest to the shortest side is within ...
0
votes
1answer
158 views

Expected length of the shortest polygonal chain connecting N random points in the unit square

N points are selected uniformly at random in the unit square. Let L(N) be the expected length of the shortest (possibly self-intersecting) polygonal chain connecting all the points. It can be proved ...
15
votes
1answer
334 views

Does every convex polyhedron have a combinatorially isomorphic counterpart whose all faces have rational areas?

Does every convex polyhedron have a combinatorially isomorphic counterpart whose faces all have rational areas? Does every convex polyhedron have a combinatorially isomorphic counterpart whose edges ...