Deprecated; please do NOT use this tag, use a more specific tag.

**8**

votes

**2**answers

4k 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 ...

**2**

votes

**2**answers

363 views

### Infinite knot composed of parallel helices

I am wondering if there is a developed theory of "infinite knots" that could capture
this object, and tell me something of its knot properties.
Imagine vertical helices in $\mathbb{R}^3$, each ...

**3**

votes

**2**answers

308 views

### Empty lattice simplex or White's theorem

White has proved (White, G. K. Lattice tetrahedra -- Canad. J. Math. 16 1964 389–396.) the following theorem:
If $T$ is a closed tetrahedron and $\Lambda$ is a lattice which contains the vertices of $...

**9**

votes

**3**answers

509 views

### Circle-arc number of a knot

I would like to build knots in $\mathbb{R}^3$ from arcs
of unit-radius (planar) circles, joined together at points where
the tangents match. Thus the knot will have
curvature $1$ at all but the ...

**2**

votes

**2**answers

540 views

### Euclidean triangulation of the plane with degree 7 at each vertex.

Hyperbolic plane has a beautiful triangulation by congruent hyperbolic triangles where all the vertices of the triangulation have degree 7, this is of course not possible in the euclidean plane, even ...

**7**

votes

**1**answer

755 views

### A remark by Gromov on 4-manifolds

Gromov remarks in a a survey on manifolds (p.12) that "it is hard to imagine that there are infinitely many non-diffeomorphic, but mutually homeomorphic, quotients of the hyperbolic 4-space by ...

**2**

votes

**0**answers

189 views

### Variations of the mean curvature

Good evening everyone,
I am facing a technical problem, maybe one of you can help.
Given a spacelike surface $S$ with mean curvature 0 in a lorentzian $3$-manifold with constant sectionnal curvature ...

**3**

votes

**1**answer

435 views

### How to partition a graph into N groups with M elements nearest?

Hi.
This problem probably already here but I could not find the right words to find it.
I have a list with 1700 points (geographic coordinates) and a need to separate into 17 groups with 100 nearest....

**9**

votes

**5**answers

551 views

### Minimal blocking objects with shadows like a cube

This is a more geometric version of the previous question,
"Lattice-cube minimal blocking sets". I will first specialize to $\mathbb{R}^3$, $d=3$.
View an $n \times n \times n$ cube $C_3(n)$ as ...

**3**

votes

**1**answer

171 views

### Are faces of a compact, convex body “opposed” iff their extreme points are pairwise “opposed”?

Let $P$ be a compact, convex subset of $\mathbb{R}^n$ (infinite-dimensional generalisations welcome, but not necessary). Let's say that disjoint subsets $W_1$, $W_2$ $\subset P$ are opposed if there ...

**5**

votes

**2**answers

288 views

### Lattice-cube minimal blocking sets

Let $C_d(n)$ be the lattice cube consisting of the $n^d$ points with
each of its $d$ coorindates in $\lbrace 1,2,\ldots,n \rbrace$.
Define a blocking set for a lattice cube to be a set of points
in ...

**2**

votes

**1**answer

143 views

### coarser than triangulations “almost partitions” into simplices

The total space $T$ of an embedded into $\mathbb{R}^n$ pure $n$-dimensional simplicial complex (in other words, the union of finitely many $n$-dimensional compact convex polytopes) sometimes admits an ...

**10**

votes

**1**answer

645 views

### A strange question about closed geodesics on a closed manifold

I'm studying a particular kind of curve evolution on Riemannian manifolds. It would help me
to know the answer to the following kinda weird question:
Does there exist a closed Riemannian manifold $M$ ...

**21**

votes

**1**answer

1k views

### A Weak Form of Borsuk's Conjecture

Problem: Let P be a d-dimensional polytope with n facets. Is it always true that P can be covered by n sets of smaller diameter?
Background and motivation
The Borsuk conjecture (disproved in 1993) ...

**3**

votes

**1**answer

312 views

### Is list of distances defining points uniquely ?

There is N points on a plane. Is is feasible to reproduce there relative location
having only list of distances. Assuming that translation, rotation and mirror are allowed
in the result. The list ...

**10**

votes

**2**answers

1k views

### Altitudes of a triangle

The three altitudes of a triangle are concurrent -- this is true in all three constant curvature geometries (Euclidean, hyperbolic, spherical), but, as far as I know, the proofs are different in the ...

**6**

votes

**3**answers

1k views

### Are properties of geodesics on a cylinder unique to cylinders?

The geodesics on a cylinder (a cylinder infinite in both directions) are either
(1) simple (non-self-intersecting) closed geodesics, or
(2) simple infinitely long geodesics (infinite in both ...

**6**

votes

**2**answers

219 views

### Number of neigbour Voronoi cells for a random set of points on S^k or cube [-1, 1]^k?

Consider $S^k \subset R^{k+1} $. Sample $N$ points by say uniform distribution. (Example k=120, N=2^24, i.e. N>>k ).
Consider Voronoi cell around each point.
How many neighbours would a cell have ...

**2**

votes

**2**answers

536 views

### Midpoint lattice polygons

Midpoint polygons (a.k.a Kasner polygons) have been studied, and their behavior is well understood.
I am considering a variant, which I call midpoint lattice polygons.
Start with a sequence of ...

**17**

votes

**1**answer

597 views

### Is the ball reducible in some high dimension?

Let $K$ be a bounded symmetric ($-K=K$) open convex body in $\mathbb R^n$. The critical determinant $d(K)$ of $K$ is the least possible volume $|\operatorname{det}(a_1\dots a_n)|$
of the fundamental ...

**11**

votes

**2**answers

671 views

### For what metrics are circles solutions of the isoperimetric problem?

A classical result is that solutions of the isoperimetric problem on the plane, the sphere, and the hyperbolic plane are circles. Are there any other Riemannian metrics on these spaces that share this ...

**1**

vote

**0**answers

339 views

### Simple development of simple curve on a cone

Let $\Lambda$ be a cone with apex $a$ and apex angle $\alpha$. Draw a simple (non-self-intersecting)
curve $C=(x,y)$ on $\Lambda$, and then develop it to a curve
$\overline{C}$ on a plane by rolling $...

**2**

votes

**1**answer

116 views

### Graph implemened into the plane with segments as edges and we search for matching with no edges intersecting

There are some points in the plane and some of them are connected with segments between them. We look at this structure as a graph implemented into the plane where the points are the vertices and the ...

**0**

votes

**5**answers

681 views

### Generate points of a (n-2)-sphere on a n-hyperplane [duplicate]

Possible Duplicate:
Efficiently sampling points uniformly from the surface of an n-sphere
I'm trying to generate random points of a (n-2)-sphere on a n-hyperplane so basically the intersection ...

**0**

votes

**1**answer

173 views

### relation with jacobifields in a small neighbourhood

hi,
I have the following question: Let $(M,g)$ be a complete Riemannian manifold with all sectional curvatures non-positive. Let $p \in M$ and consider the function $d(x)=dist_{g}(x,p)$ in a ...

**5**

votes

**1**answer

320 views

### Geometric applications of Ekeland's variational principle

I'm looking for geometric applications of Ekeland's variational principle in order to see it at work in a context I'm familiar with. Let me recall the principle itself:
Definition. Let $(X,d)$ be a ...

**3**

votes

**0**answers

69 views

### Node-Weighted Euclidean Steiner Trees

I would like to know whether the following problem, including algorithms to solve it (exact or approximations) has been studied.
A finite set of positive-weighted points are given in the n-...

**4**

votes

**1**answer

277 views

### Is it possible to sample uniformly on the surface of a high-dimensional polytope?

There are some pretty simple methods to do uniform sampling on the surface of high-dimensional spheres or cubes.
Are there any methods that sample uniformly on the surface of a high-dimensional ...

**16**

votes

**3**answers

1k views

### Prescribing areas of parallelograms (or 2x2 principal minors)

Let $(a_{ij})$ be a $n\times n$ symmetric matrix such that $a_{ij}\geq 0$ for all $i,j$ and $a_{ii}=0$ for all $i$. Under which conditions on the $a_{ij}$'s can one find $n$ vectors $v_1,\ldots,v_n\in{...

**5**

votes

**2**answers

595 views

### Pursuit-Evasion on a Manifold

I know pursuit-evasion has been studied in many contexts, including
on a manifold (e.g., Melikyan,
"Geometry of Pursuit-Evasion Games on Two-Dimensional Manifolds"),
but I have not seen this version:
...

**9**

votes

**1**answer

1k views

### What nets fold to polyhedra?

There is a classic (and open) problem asking whether every polyhedron can be unfolded to give a non-overlapping net. The converse problem has been studied asking which polygons can be folded in some ...

**14**

votes

**5**answers

981 views

### How far is a set of vectors from being orthogonal?

Given some vectors, how many dimensions do you need to add (to their span) before you can find some mutually orthogonal vectors that project down to the original ones?
Or, more formally...
Suppose $...

**9**

votes

**3**answers

516 views

### Recognising group actions on trees from the boundary

Let $G$ be a group acting on a locally finite tree $T$. Then the boundary $\partial T$ is a Cantor set on which $G$ acts by homeomorphisms (indeed by quasi-isometries under a suitable metric). ...

**4**

votes

**2**answers

503 views

### Real vs complex surfaces

Hi! My background is in (complex) algebraic geometry. For the first time I'm considering varieties defined over $\mathbb{R}$ and I have some very basic questions about these.
In particular I'm trying ...

**5**

votes

**2**answers

232 views

### Normal form(s) for the elements of hyperbolic triangle groups

I'm seeking a reference or a sketch for any sort of normal form that would enable rapid enumeration without redundancies of the elements of hyperbolic triangle groups and/or von Dyck groups.

**4**

votes

**1**answer

722 views

### Polyline Averaging

I'm trying to find a method that can take a collection of polylines, each described by a list of connected points on a plane, and find an "average" path through them. The input lines do not loop.
...

**6**

votes

**0**answers

351 views

### Maximum volume convex body coverable by a unit square

Suppose you are given a single unit square, and you are permitted to cut it into $k$ (connected)
pieces (where $k=1$ means just the square). Your task is to construct the largest volume
convex body ...

**13**

votes

**3**answers

1k views

### Covering a Cube with a Square

Suppose you are given a single unit square, and you would like to completely cover the surface
of a cube by cutting up the square and pasting it onto the cube's surface.
Q1. What is the largest ...

**5**

votes

**2**answers

414 views

### Tangled Knot Function

I am seeking a function $f: \mathbb{R}^3 \to \mathbb{R}^3$
that has these properties:
(1) When iterated $n$ times starting from some $p$,
connecting the points in order
with segments and closing ...

**2**

votes

**2**answers

300 views

### Questions on calculating volume using n-1 forms

Is there an n-1 form on $R^n$ which calculates the volume of n-manifolds? Similarly, is there such a 1 form on $S^2$, and $RP^2$? I thought this has something to do with the orientation, is that right?...

**12**

votes

**2**answers

576 views

### What is the largest possible thirteenth kissing sphere?

It is well-known that it is impossible to arrange 13 spheres of unit radius all tangent to another unit sphere without their interiors intersecting. This was apparently the subject of disagreement ...

**5**

votes

**3**answers

488 views

### Tetrahedron angles sum to $\pi$: Bisector plane

I discovered empirically what to me is an amazing lemma
concerning face angles of a tetrahedron.
Let $\triangle abc$ be a triangle in the $xy$-plane, and $d$ the
apex of a tetrahedron with positive $z$...

**10**

votes

**3**answers

848 views

### Optimal Wireframe Sphere

Suppose you have a length $L$ of metal pipe at your disposal,
and you would like to build a wireframe unit-radius sphere,
by bending, cutting, and welding the pipe into a connected structure $F$.
Your ...

**5**

votes

**1**answer

266 views

### Checking if one polytope is contained in another

Hi,
I have two sets of inequalities, say, $Ax \leq 0$ and $Bx \leq 0$. I would like to know if they both define the same polytope. Or, even, whether one is contained in the other.
At the moment I am ...

**1**

vote

**2**answers

3k views

### Find the bounding box of a circle segment.

You have three points. A,B and C.
They define a circle segment that starts at A, goes through B and ends at C.
Find the smallest bounding box that encompases the circle segment.
Here is a picture:
...

**3**

votes

**2**answers

783 views

### Maximum area of intersection between annulus and circle? [closed]

Given two concentric circles $[C_1,C_2 ]$ with radii $(R_1 < R_2) $ creating an annulus; where should a third circle ($C_3$, radius $R_3$) be located such that the area of intersection between ...

**4**

votes

**2**answers

595 views

### Area ratio of a minimum bounding rectangle of a convex polygon

Take a convex polygon $P$ in the plane, and find its minimum area bounding rectangle, $R$. I'm interested in the ratio of the area of $R$ to the area of $P$. The ratio has a minimum of $1$ for ...

**7**

votes

**1**answer

355 views

### Generalizing a square wheel to a body rolling on a surface

A square wheel rolling on a catenary road maintains the wheel center at a fixed
height, a well-known construction previously discussed on MO
(e.g.,
"Generalizing square wheels rolling on inverted ...

**8**

votes

**4**answers

878 views

### Geodesic path on the unit sphere with the sup norm

Let $X$ be the unit sphere of $\mathbb{R}^n$ with the sup norm, i.e. $X=\{x\in\mathbb{R}^n: \|x\|_{\infty}=1\}$. Let the metric $d$ on $X$ be the geodesic metric induced by the sup norm, i.e for any ...

**2**

votes

**1**answer

518 views

### Filling $\mathbb{R}^3$ with skew lines

I would like to know if it is possible to fill $\mathbb{R}^3$ with lines with the
following two properties:
(1) Every point $x \in \mathbb{R}^3$ is contained in precisely one line.
(2) Every ...