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8
votes
2answers
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
2answers
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
2answers
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
3answers
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
2answers
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
1answer
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
0answers
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
1answer
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
5answers
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
1answer
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
2answers
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
1answer
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
1answer
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
1answer
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
1answer
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
2answers
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
3answers
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
2answers
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
2answers
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
1answer
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
2answers
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
0answers
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
1answer
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
5answers
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
1answer
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
1answer
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
0answers
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
1answer
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
3answers
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
2answers
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
1answer
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
5answers
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
3answers
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
2answers
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
2answers
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
1answer
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
0answers
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
3answers
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
2answers
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
2answers
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
2answers
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
3answers
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
3answers
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
1answer
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
2answers
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
2answers
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
2answers
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
1answer
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
4answers
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
1answer
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 ...