**4**

votes

**0**answers

120 views

### Graph drawing maximizing the volume of the convex hull

Given a graph $G=(V,E)$ and a length function $\ell:E\to\mathbb{R}_+$.
An embedding of the graph into the $d$-dimensional Euclidean space is a map $f:V\to\mathbb{R}^d$ such that ...

**3**

votes

**2**answers

298 views

### Questions on Discrete Exterior Calculus in numerial computing

I have several questions about the Discrete Exterior Calculus (DEC) in the numerical method for solving partial differential equation in physics:
(Discrete Exterious Calculus is the newly developed ...

**2**

votes

**2**answers

209 views

### Primitive orthogonal vectors/Unimodular matrices

Primitive sets of vectors are very important in the theory of point lattices, since they constitute the sets of vectors that are part of a basis for the lattice.
A set of integer vectors ...

**11**

votes

**2**answers

774 views

### Access to a preprint by D. N. Verma

Some work I am doing is connected with a sequence 1, 3, 40, 1225, 67956, $\dots$ which agrees with http://oeis.org/A012250 for all eight terms. The only useful information in OEIS on this sequence is ...

**6**

votes

**2**answers

172 views

### Untangling entwined rigid chains in 3-space

I am interested in exploring the degree of "tangledness"
of two rigid chains in space.
A polygonal chain is a simple (non-self-intersecting) path
of segments in
$\mathbb{R}^3$, viewed as a rigid body. ...

**3**

votes

**1**answer

147 views

### Triangulation of the surface determined by sampling two of its cross-sections

I have a data set that essentially looks like the picture below, i.e., it's given by sets of points in $\mathbb{R}^3$ that sample the cross-sections of a certain surface that in principle I do not ...

**7**

votes

**0**answers

141 views

### Fractional Helly for more than one piercing

Fractional Helly Theorem says the following:
For every $0<\alpha\leq 1$ there exists $\beta = \beta(d, \alpha)$ with the following property. Let $C_1 , C_2 , ..., C_n$ be convex sets in $R^d$, $n ...

**3**

votes

**1**answer

203 views

### What properties does generalized Delaunay triangulation have?

Suppose that instead of the usual circle, we pick some other convex set D and make the Delaunay triangulation of a finite planar point set with respect to this set, i.e. connect two points if there is ...

**1**

vote

**1**answer

300 views

### Finding a point farthest away from $k$ points in a polygon

There are $k$ points placed inside a polygon and I am interested in finding a point inside the polygon (not necessarily on its boundary) who's minimum distance to any of the $k$ points is maximized.
...

**2**

votes

**1**answer

243 views

### Ways to look at a polyhedral graph

Motivation
There are at least three interpretations of an abstract polyhedral (= planar 3-vertex-connected) graph:
the 1-skeleton of a convex polyhedron (when embedded into $\mathbb{R}^3$)
a ...

**12**

votes

**0**answers

422 views

### Does every connected set that is not a line segment cross some dyadic square?

A dyadic square is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists a connected subset ...

**8**

votes

**0**answers

119 views

### Diameter of simplicial complex mirrored in property of Stanley-Reisner ring?

Consider a pure finite abstract simplicial complex $\Delta$. Define its diameter as the maximal distance between any two facets, i.e., between any two faces of maximal dimension $d-1$. The distance ...

**18**

votes

**3**answers

1k views

### Research trends in geometry of numbers?

Geometry of numbers was initiated by Hermann Minkowski roughly a hundred years ago. At its heart is the relation between lattices (the group, not the poset) and convex bodies. One of its fundamental ...

**2**

votes

**1**answer

329 views

### Apollonian gasket and the degree of convergence

Let $r_1,r_2\dots$ be the radii of Apollonian gasket.
I would like to know for which values $\alpha$ we have
$$\sum_{n=1}^\infty r_n^\alpha<\infty.$$
I know that if three circles $A$, $B$ and ...

**1**

vote

**0**answers

122 views

### Spectrum of Combinatorial Laplacian

The spectrum of the combinatorial laplacian is well understood for a square lattice. What about for other lattices?
In particular:
Let $ f: \mathbb{Z}^2 \rightarrow \mathbb{R} $. The usual ...

**2**

votes

**2**answers

215 views

### Is there a combinatorial analogue of the Kazdan Warner theorem?

First let me state a result of Kazdan and Warner
Let $M$ be a compact orientable two dimensional manifold.
Let $f:M \rightarrow \mathbb{R}$ be a function that has the same
sign as the Euler ...

**25**

votes

**3**answers

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 $K$ ...

**23**

votes

**2**answers

779 views

### The kissing number of a square, cube, hypercube?

How many nonoverlapping unit squares can (nonoverlappingly) touch one unit square?
By "nonoverlapping" I mean: not sharing an interior point.
By "touch" I mean: sharing a boundary point.
...

**8**

votes

**2**answers

558 views

### A Problem about partitioning $S^2$

Question: Can the 2-dimensional sphere $S^2$ be partitioned into four nonempty sets such that every circle in $S^2$ passes through just three of these four sets?
Here, "just three" means "exactly ...

**21**

votes

**1**answer

682 views

### Sane bound on number of moves for Maker-Breaker game on $\mathbb R^2$ for $\{0,1,2,3,4\}$

The description below comes from
József Beck. Combinatorial games. Tic-tac-toe theory, Encyclopedia of Mathematics and its Applications, 114. Cambridge University Press, Cambridge, 2008, MR2402857 ...

**2**

votes

**2**answers

252 views

### packing disks tightly in the plane

Given a discrete point set $S$ in ${\bf R}^2$ with a specified base-point $p_0 \in S$, label the remaining points as $p_1, p_2, \dots$ in order of increasing distance from $p_0$ (with ties resolved ...

**2**

votes

**2**answers

227 views

### What is the smallest number of subsets in such a subdivision?

Given any $30$ points in the plane, what is the smallest number of
subsets in a subdivision of the set of $30$ points into subsets such
that all the points in each subset are on the boundary of the ...

**4**

votes

**1**answer

411 views

### Combinatorial geodesics

[There has been a flaw in my definition - as Sergei and Andreas pointed out. I hope I could fix it.]
I want to understand how the concepts of directions, straight (or shortest) lines, and geodesics ...

**8**

votes

**1**answer

388 views

### Polyhedra that combinatorially shadow a sequence

Let $P$ be a polyhedron in $\mathbb{R}^3$.
Say that $P$ combinatorially shadows a sequence of natural numbers $S$ if
there is a continuous rotation of $P$ such that its orthogonal-projection
shadows ...

**0**

votes

**1**answer

215 views

### Is this bounded?

May be better to ask for help here. Let $v_{1}$, $v_{2}$, $\ldots$, $v_{m}$ be the vertices of a
convex polygon in the plane and $v_{m+1}$ be a vertex in the interior
of the convex polygon. Connect ...

**9**

votes

**2**answers

343 views

### On well separated point sets in the plane

Let us say that a finite set $A$ in the plane is $1$-separated if:
1) it has an even number of points;
2) no open ball of diameter $1$ contains more than $|A|/2$ points.
For a $1$-separated set $A$ ...

**14**

votes

**1**answer

427 views

### The optimal constant in Vitali covering lemma

Let me restate Vitali covering lemma.
Let $\{B_i\}_{i\in F}$ be a finite collection of balls in the $\mathbb{R}^n$. Then there is $S\subset F$ such that the balls $\{B_i\}_{i\in S}$ are disjoint and
...

**14**

votes

**2**answers

716 views

### Particles chasing one another around a circle

Two particles start out at random positions on a unit-circumference circle.
Each has a random speed (distance per unit time) moving counterclockwise uniformly distributed
within $[0,1]$. How long ...

**1**

vote

**0**answers

165 views

### Lattice-point enumeration question involving linear combinations of matrices

I would like to know some references to learn more about an answer to this question, if there are any references:
Let $A_1, \dots , A_m$ and $B$ be $n\times n$ symmetric matrices. Let $$S = \{(x_1, ...

**2**

votes

**0**answers

98 views

### Rational viewing points in a polygon

We refer to the question posed in Seeing the vertices of a polygon with rational angles, but now ask for constructions or for the existence of rational viewing points. We'll call a point $p$ inside ...

**6**

votes

**1**answer

196 views

### Seeing the vertices of a polygon with rational angles

Given any convex polygon in the plane, is it always possible to find a point $p$ in its interior such that when we draw the line segments from $p$ to each of its vertices, the angles formed at $p$ are ...

**13**

votes

**1**answer

325 views

### The sparsest planar net that captures every unit segment

Let $\cal C = \lbrace C_i \rbrace$ be a collection
of rectifiable curves in the plane with the property that
every unit-length segment meets at least one curve
in at least one point.
Call such a ...

**11**

votes

**0**answers

379 views

### Reciprocity (Ehrhart-style) for real polytopes?

Is there some sense in which the well-known Ehrhart reciprocity law for rational, convex, polytopes can be extended to any convex polytope with arbitrary real vertices?
In other words, given any ...

**12**

votes

**1**answer

571 views

### The geometry of crinkled aluminum foil

I wonder if the geometry of crinkled aluminum foil has been studied?
The above is a photo of foil I flattened to reuse.
It might be ...

**4**

votes

**1**answer

246 views

### Best upper bound on rate for 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 which states that the rate $R(\delta)$ corresponding to ...

**0**

votes

**2**answers

135 views

### Disks Packing Variant

Usually disk packing problems require that no two disks of the packing intersect.
Does anybody know if the problem has been studied when disks may intersect but they are not allowed to contain the ...

**37**

votes

**8**answers

3k views

### A sudden smiley? :-)

This is a vague question, and I will no doubt be (properly!) chastised for posing it.
I would like to generate a set $S$ of points in $\mathbb{R}^3$—$|S|$ finite or infinite—which
has the ...

**2**

votes

**2**answers

189 views

### Is there a simple test to determine whether a polytope is integral?

It is known that any rational convex polytope expressed as $\{ x\in\mathbb{R}^d : Ax \ge b \}$, where $A\in\mathbb{Z}^{k\times d}$ and $b\in\mathbb{Z}^k$, can be written as the convex hull of finitely ...

**9**

votes

**2**answers

545 views

### Discrete Morse function from smooth one

Suppose I have a smooth manifold $M$ and a smooth Morse function $f$ on $M$. Is there a standard way to replace $M$ with a cell complex and $f$ with a discrete Morse function such the resulting ...

**0**

votes

**3**answers

277 views

### The symmetry group of $\mathbb Z^d$

Let $d \ge 1$, and consider the integer lattice $\mathbb Z^d$. This is a homogeneous space, in the manner of the Erlangan Programm.
I would like to write $\mathbb Z^d = G / H$, where $G$ is the ...

**5**

votes

**0**answers

456 views

### Optimal Gear Trains

Suppose you need to slow down a turning motor so that a gear turns at
an angular velocity $\frac{a}{b}$ of that of the motor shaft, where $a$ and
$b$ are natural numbers. For example, this set of ...

**6**

votes

**3**answers

343 views

### Herringbone partitions of regions and surfaces

Let $R \subset \mathbb{R}^2$ be a region of the plane bounded
by a Jordan curve. The boundary $\partial R$ could be a polygon,
or a smooth curve—there are variations depending upon boundary ...

**3**

votes

**1**answer

223 views

### Empty convex polytopes for random point sets

I know of the famous results on the Erdős-Szekeres empty convex polygon problem in the plane
(the Happy-Ending Problem), and I know that there are higher-dimensional extensions.
A great source ...

**4**

votes

**2**answers

301 views

### Realization spaces for regular convex polytopes

Q1.
Are there convex polytopes combinatorially equivalent to each of the regular polytopes
that are realized with integer vertex coordinates?
...

**11**

votes

**1**answer

332 views

### Detecting a hidden convex body with line probes

Imagine that, somewhere inside an origin-centered, unit-radius sphere
$S$ in $\mathbb{R}^3$,
sits a convex body $K$ of volume vol$(K)=\alpha (\frac{4}{3} \pi)$,
with $\alpha < 1$ the fraction of ...

**7**

votes

**1**answer

2k views

### Get Largest Inscribed Rectangle of a Concave Polygon

I'm looking for an algorithm to find a set of largest inscribed rectangles of a concave polygon where each rectangle must be collinear with one of the edges of the polygon.
In other words, I want to ...

**8**

votes

**2**answers

2k views

### Covering a Polygon with Rectangles

I am tyring to cover a simple concave polygon with a minimum rectangles. My rectangles can be any length, but they have maximum widths, and the polygon will never have an acute angle.
I thought about ...

**1**

vote

**2**answers

197 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?

**5**

votes

**0**answers

312 views

### $n$ lines in a general position and the number of empty triangles

Question. Consider $n \geq 5$ lines in a general position (i.e. no two lines are parallel and no triple intersections are allowed) in $\mathbb{R}^2$. Let $T(n)$ denote the maximal number of empty ...

**3**

votes

**2**answers

355 views

### What collections of convex sets result in non-trivial uses of Helly's theorem?

Consider Helly Theorem, taken from notes by Igor Pak:
Let $X_1, \dots, X_n \in {\mathbb{R}}^2$ be convex regions in the plane such that any triple interesects $X_i \cap X_j \cap X_k \neq 0$. Then ...