Deprecated; please use a more specific tag.

**0**

votes

**1**answer

366 views

### Action of Isometries on a Line in the Plane

I'm trying to determine the stabilizer of a line in a plane when acted upon by the group of isometries of the plane. Please note that I'm using the notation found in the Wikipedia article on Euclidean ...

**0**

votes

**2**answers

240 views

### Derivation of surfaces

In an Euclidean linear plane, the evolute of a given curve C with support function h(t) can be regarded as a kind of derivative C' of C. Indeed, C' has support function h'(Pi/2 -t).
Is there any ...

**3**

votes

**0**answers

128 views

### The mean number of vertices in small connected components of random geometric graphs

I place $N$ points on a circular plane of radius $R$, and draw edges to connect points that are less than or equal to some distance $D$ to form a set of graphs or cliques $G_i$. As a function of $N$, ...

**1**

vote

**1**answer

235 views

### A raceway problem

Let $f(x)=\sin x$, and $g(x)=\sin x + 1$. Consider a set
$S=\{(x,y)| f(x)\leq y \leq g(x), x\in [0,2\pi]\}$. This set $S$ can be considered as "Raceway"
My question is finding the shortest path in ...

**6**

votes

**2**answers

724 views

### Finding the convex combination of vertices which yields an inner point of a polytope

Given a convex polytope $P\in \mathbb{R}^n$, and a point $x\in P$, Caratheodory's theorem gives us that there exists a set of at most $n+1$ vertices of $P$, such that $x$ is a convex combination of ...

**10**

votes

**1**answer

535 views

### Polygons uniquely inducing arrangements

A beautiful, relatively recent result is that,
Every simple arrangement $\cal{A}$ of $n$ lines in the plane is induced by a simple $n$-gon $P$.
In a simple arrangement, every pair of lines ...

**9**

votes

**2**answers

583 views

### Ellipsoids and lattices: an enclosure problem.

$E \subset {\mathbb R}^2$ is an ellipse of area $1$ centered at the origin that contains no other point with integer coordinates. Is there a matrix $A \in SL(2,{\mathbb Z})$ such that the ellipse ...

**3**

votes

**4**answers

1k views

### space of geodesics

hallo,
i have the following problem: Let $(M,g)$ be a compact Riemannian manifold with metric $g$ and $\nabla$ be the Levi-Civita Connection. Denote by $G(M) =${$\gamma: \mathbb{R} \rightarrow M | ...

**2**

votes

**1**answer

199 views

### Locus of points where difference in gravitational forces is constant

Is there a name for the curve in the plane defined by
$a/\|x - p\|^2 - b/\|x - q\|^2=\mathrm{constant}$
where $a$ and $b$ are fixed numbers and $p$ and $q$ are fixed points? How about if I don't ...

**9**

votes

**4**answers

429 views

### Can one do without a classifying space when showing vanishing of cohomology

Let $G$ be a discrete group and $A$ an abelian group, then $H^n (G,A)$ can be defined as $$ H^n (G,A) = H^n (B_G, A)$$
Where $B_G$ is the classifying space of $G$, i.e. $B_G = E_G / G$ where $E_G$ is ...

**3**

votes

**2**answers

766 views

### absolute continuity on $R^{n}$

I know the definition of absolute continuity if there is a function $f:(a,b)\rightarrow R$.
I wonder what is an analogy of this concept if we have a function $f:A\rightarrow R$, where $A\subset R^{n}$ ...

**0**

votes

**1**answer

2k views

### Tangent lines to 2 circles, tangent planes to 3 spheres, and so on.

Although it is known the solution to the first two questions, somebody may have different nice answers, so I include them:
Given two circles in the plane, there is (at least) a line which is tangent ...

**13**

votes

**1**answer

550 views

### Ratio of circumscribed/inscribed $(n{-}1)$-gons

As a discrete analog of the MO question,
"Löwner-John Ellipsoid: incribed and circumscribed,"
I've been wondering what might be the maximum ratio
of this quantity?
Let $P$ be a convex polygon of $n$ ...

**-1**

votes

**2**answers

537 views

### projective camera: back-projecting a point on the image plane into 3-space

suppose I got a projective camera model. for this model I would like to back-project a ray through a point in the image plane. I know that the equation for this is the following:
$$
y(\lambda) = P^+_0 ...

**5**

votes

**0**answers

272 views

### Local minimum from directional derivatives in the space of convex bodies

I have a function $f(K)$ defined on the space of three-dimensional convex bodies for which I want to show that the unit ball $B$ is a local minimum. I have been able to show if $K$ is not homothetic ...

**5**

votes

**1**answer

464 views

### Bending an elastic, inextensible sheet of paper into a teardrop shape

I take an rectangular sheet of paper, of height $H$ and Young's Modulus $E$, and in the absence of gravity, I bend it into a "teardrop" shape so that the edges along the top and bottom touch only ...

**4**

votes

**0**answers

90 views

### Applications of k-medians with moment constraints

Suppose I have a collection of points $p_1,\dots,p_N$ in the plane. In the Euclidean $k$-medians (or $k$-means) problems, our objective is to distribute a set of $k$ points $x_1,\dots,x_k$ in the ...

**10**

votes

**2**answers

471 views

### The “grassmannian” of a simplicial complex

This question is mainly a reference request – I have a construction which seems natural, so I am quite convinced it should be standard, but I don't know what it is called.
Take an $n$ dimensional ...

**1**

vote

**0**answers

114 views

### Proving that an optimal solution “converges”

This question is a follow-up on a previous question I asked at:
Distances between and among points in a region
Let $X = x_1,\dots,x_n$ denote a finite set of $n$ points in the unit circle $C$ in the ...

**0**

votes

**1**answer

217 views

### Shortest distance along the surface of the hyperboloid [closed]

Given two points A and B on the surface of the hyperboloid x^2+y^2-z^2=1. How to find the shortest distance between them along the surface?

**5**

votes

**1**answer

192 views

### Reference for a dual isoperimetric problem and solution

I am trying to track down the first published solution to the following problem:
What curves within the unit disc in the plane and endpoints on the unit circle, minimize their length (within the ...

**7**

votes

**2**answers

505 views

### Fitting a mesh to a density function

Suppose I have a probability density function defined on a region in the plane (in my case, the pdf is of the form $f(x) = \alpha\|x\|^{-\beta}$, and the region is the unit disk). For large $N$, is ...

**6**

votes

**3**answers

1k views

### Original proof of Pappus' Hexagon Theorem

Does anyone know where I can find an english translation, preferrably online or in a book the library of a small liberal arts college would be likely to have, of the original proof of Pappus' hexagon ...

**5**

votes

**0**answers

277 views

### degenerating surface II

In degenerating surface, Robert Bryant give us an example of a sequence of minimal immersions which converges (in $C^2$- topology) to $z\mapsto z^{2k+1}$ on the unit disc $\mathbb{D}$. My question is ...

**5**

votes

**1**answer

1k views

### Good Surface,Bad Surface-Surface classification

Maybe this question be very simple, but I don't know why it is hard for me. Thanks for any guide and help.
We say a surface $S$ (2-dimensional metric(compact) Riemannian surface) is good (denote by ...

**10**

votes

**2**answers

473 views

### Periodic lightray paths trapped between two nested mirror circles

I wonder if the periodic paths of a lightray trapped between two nonconcentric circles,
each perfectly reflecting, are known?
The behavior of such rays seems chaotically complicated. For example, ...

**1**

vote

**1**answer

401 views

### Program for drawing cobordisms [closed]

Perhaps this is not the right place to ask the following question but I did not find any suitable on the web. So I would be very grateful for sharing your experience.
What is a good program to draw ...

**4**

votes

**2**answers

895 views

### Selecting two random points inside a sphere which are a fixed distance apart

Without appealing to a guess-and-check approach, how might I select a pair of random points inside of a sphere of radius $R$ s.t. the points always a distance $d \leq R$ apart? Can the selected ...

**0**

votes

**1**answer

362 views

### Minimum distance between two data sets

Suppose we have two sets of data, $X$ and $Y$, each of which contains
$10$ positive numbers. Now let us order the data sets $X=\left\{ x_{1},\cdots,x_{10}\right\}$,
$x_{1}\ge\cdots\ge x_{10}>0$ and ...

**6**

votes

**3**answers

1k views

### Why are isometries of Minkowski space necessarily linear?

The Mazur-Ulam theorem says that any surjective isometry of normed vector spaces is affine. This argument doesn't seem to apply to Minkowski space (of special relativity) since the metric is ...

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votes

**2**answers

3k views

### Best fit circles inside a square based on side of the square [closed]

Hi,
How to calculate number of circles in side a square, if we know the side of the square and the circles all are equal size.
Thanks.

**6**

votes

**2**answers

684 views

### Computational complexity of integration in two dimensions

I have an algorithm for solving a certain problem that requires that I compute two-dimensional integrals as a subroutine, and I'd like to make some kind of statement about its running time. Suppose ...

**0**

votes

**1**answer

328 views

### existence of fractal [closed]

I have a question about fractals;
Suppose $\alpha\in[0,1]$ is real number, is there any fractal $F_\alpha$, such that $Dimension(F_\alpha)=\alpha$?
If yes, do we have any method to construct such ...

**3**

votes

**1**answer

272 views

### Radial tilings with variable area ratios

I was looking at this neat page on logarithmic spiral tilings when a question popped up:
http://www.uwgb.edu/dutchs/symmetry/log-spir.htm
It seems that in all of the tilings shown, the area of each ...

**2**

votes

**1**answer

231 views

### Minimum distance to a sampled point with given pdf

Let $f(x)>0$ be a probability density function defined on the unit square $[0,1]^2$ in $\mathbb{R}^2$. Suppose that we take $N$ independent samples, $X_1,\dots,X_N$, of $f$. Now, sample a point ...

**17**

votes

**2**answers

813 views

### Placing points on a sphere so that no 3 lie close to the same plane

Motivation
I am working with arbitrary parallelopiped tilings given by projection from a higher dimensional space. The collection of tiles, and some properties of the higher dimensional space are ...

**8**

votes

**2**answers

449 views

### A set of points congruent to its proper subset

There are sets of points in $\mathbb{R}^n$ congruent to their own proper subsets. A (trivial) example is a ray, or to give a more interesting bounded example, $\{e^{i \cdot n} \mid n\in\mathbb{N}\}$. ...

**11**

votes

**2**answers

558 views

### Maximum thickness of three linked Euclidean solid tori

Consider three circles of radius 1 in $\mathbb{R}^3$, linked with each other in the same arrangement as three fibers of the Hopf fibration. Now thicken the circles up into non-overlapping standard ...

**4**

votes

**2**answers

209 views

### What is the smallest diameter ring a non-convex polyhedron can pass through in 3-space?

The question is mostly in the title.
Imagine I have some non-convex polyhedron $P$, and I would like to find the smallest diameter ring that it can pass through in 3-space, undergoing any necessary ...

**8**

votes

**0**answers

347 views

### is a group $G$, that admits finite $k(G, 1)$ and has no Baumslag-Solitar subgroups, necessarily hyperbolic?

This is the first question asked in Bestvina's article "Questions in Geometric Group Theory". Does anyone know if there has been any progress made on this problem? Is the question answered if $G$ is ...

**2**

votes

**0**answers

86 views

### An equilibrium property of subsets of $\mathbb{R}^{n}$

Let $A\subset\mathbb{R}^{n}$ be a closed set and $\phi(x)=e^{-||x||^{2}}$,
$x\in\mathbb{R}^{n}$. We shall say that $A$ is an "equilibrium set", if
$x\in\partial A$ ...

**2**

votes

**1**answer

362 views

### Configuration space of points in Euclidean Space with fixed distances

What can we say about the configuration space of $n$ points in $\mathbb{R}^{m}$ with fixed distances between the points ?
e.g. for 2 points in $\mathbb{R}^{3}$ with fixed distance between them,the ...

**18**

votes

**2**answers

1k views

### Probing a manifold with geodesics

Supposed you stand at a point $p \in M$ on a smooth 2-manifold $M$
embedded in $\mathbb{R}^3$.
You do not know anything about $M$.
You shoot off a geodesic $\gamma$ in some direction $u$,
and learn ...

**6**

votes

**1**answer

295 views

### Tiling by regular simplices

The plane can be tiled without gaps by congruent two-dimensional regular simplices (i.e., equilateral triangles). The three-dimensional Euclidean space cannot be tiled by congruent three-dimensional ...

**1**

vote

**0**answers

155 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

831 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 ...

**16**

votes

**0**answers

509 views

### Random Distance Matrices

My question is motivated by the following recent paper:
http://arxiv.org/abs/1110.6333
Assume you have a metric space $(X,d)$ equipped with a Borel probability measure $\mu$. We can further assume ...

**4**

votes

**2**answers

271 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

158 views

### Manifolds with a lower degree of regularity

Hello guys,
I've been reading a paper about regularity theory for a P.D.E in a non-smooth domain(see the reference below).
There, the authors consider domains of $R^n$ with regularity of class $W^2 ...

**3**

votes

**0**answers

147 views

### An isoperimetric inequality for “order” polytopes

I am looking for an isoperimetric inequality for order-like polytopes.
An order polytope $K\in \mathbb{R}^n$ is defined by a set of linear inequaities:
$$ \forall i \; 0\leq x_i \leq 1 $$
and
$ ...