All Questions
4,827 questions
0
votes
1
answer
327
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Minimum cardinality of the intersection of 2D rectangles
Let $S$ be a set of 2D points $(x,y)$ with positive real coordinates, i.e. $x,y>0$. An 2D rectangle $R$ is called an ${Origin-Rectangle}$ if it is decided by the origin $(0,0)$ and another point $(...
15
votes
2
answers
737
views
Tiling survey that updates "Tilings and patterns"?
Can anyone suggest a survey (or surveys) that provides an update to Tilings and patterns by Grunbaum and Shepard? If there's a more recent book, that would be fantastic, but I don't see one.
I am ...
- 881
20
votes
2
answers
25k
views
Partitioning a polygon into convex parts
I'm looking for an algorithm to partition any simple closed polygon into convex sub-polygons--preferably as few as possible.
I know almost nothing about this subject, so I've been searching on Google ...
- 201
3
votes
4
answers
4k
views
Existence of nonnegative solutions to an underdetermined system of linear equations
Similar questions have been asked elsewhere, but I think this is sufficiently different to warrant a new post. I have a particular matrix $A$ and would like to know when the system $Ax = 0$ has at ...
- 491
1
vote
0
answers
246
views
How to derive an energy measure of metric deforming
The problem is an abstract from applied science.
Given an $n$ dimensional Riemann manifold with metric $\langle M, g\rangle$, we could define deformation of the metric $g(t)$ where $t\in [0,1]$, for ...
- 135
7
votes
1
answer
360
views
Standard reference for equivalence of PU(2) action on $\mathbb{C}\mathbb{P}^1$ and SO(3) action on $S^2$
The equivalence I describe below is well-known, but I'd like a simple standard reference for it.
Consider $\mathbb{C}\mathbb{P}^1$, the set of one-dimensional subspaces of $\mathbb{C}^2$, which has a ...
- 2,210
5
votes
2
answers
1k
views
Find the point on the Stiefel Manifold that is closest to a matrix
I don't have much background on high-dimensional geometry, so I dare to ask it.
For a given point in $x\in\mathbb{R}^n$, assume that I want to find the point on the unit sphere that is closest to the ...
11
votes
4
answers
6k
views
Place of Analytic geometry in modern undergraduate curriculum
I am a freshmen student in mathematics at Moscow State University (in Russia) and I'm confused with placing the subject called "analytic geometry" into the system of mathematical knowledge (if you ...
- 111
13
votes
2
answers
664
views
Helix translates as geodesics
I believe one can fill $\mathbb{R}^3$ with
horizontal translates of the helix
$(\cos t, \sin t, t) \;,\; t \in \mathbb{R}$,
so that every point of $\mathbb{R}^3$
lies in exactly one helix.
I am ...
- 151k
6
votes
3
answers
11k
views
Maximum flow with negative capacities?
I'm trying to compute an (s-t) maximum flow through a network which includes a number of arc pairs ((u,v), (v,u)) that have equal, negative capacities (weights). I'm not aware of any efficient ...
- 83
16
votes
3
answers
2k
views
Expected Degree of a vertex in Delaunay Triangulations
Assume you have a Poisson point process of constant intensity $\lambda$ in the Euclidean plane. From this point process we construct the Delaunay triangulation (or the Voronoi tessellation for that ...
- 3,626
0
votes
1
answer
250
views
Hyperbolic isometries in cocompact Hadamard (i.e. cat(0) proper simply connected) spaces
Swenson proved in "A cut point theorem for ${\rm CAT}(0)$ groups" that a locally compact Hadamard space with a geometric action by a group $G$ admits a hyperbolic isometry (that lie in $G$).
Is it ...
- 19
5
votes
3
answers
464
views
Quantitative measurement of infinite dimensionality
I recently encountered the metric mean dimension, which is a numerical metric invariant of (discrete time, compact space) dynamical systems that refines topological entropy for infinite-entropy ...
- 14.4k
8
votes
2
answers
991
views
Will a ball fired through a focus of an ellipse eventually tend to a horizontal line?
A couple of years ago I came across this phenomenon which appears to be true although I am having difficulty proving it.
F and F' are foci of a billiard table in the shape of an ellipse. A ball is ...
- 83
4
votes
4
answers
703
views
efficient way to compute the inversion of the following matrix
Hi, there
I have looked it up in the current textbook. The conventional numerical method to compute the inversion of an $n \times n$ matrix requires $O(n^3)$. However, for the following special ...
- 41
1
vote
1
answer
248
views
Is there a good approximating polygon for every smooth set?
Suppose we have an open set S whose boundary is a closed Jordan curve that has a unique tangent at each point. Is it true that for every epsilon there is a P polygon contained in S such that there is ...
- 19k
2
votes
3
answers
2k
views
Are there non-measurable sets with smooth boundary?
I learned analysis a while ago, so let me define what I want. Suppose we have a set whose boundary is a closed Jordan curve that has a unique tangent at each point. Is it true that this set is (...
- 19k
7
votes
0
answers
208
views
How do metrics behave under joining along a manifold embedded in the boundary?
How do metrics behave under joining along a manifold embedded in the boundary?
This is, more-or-less, part of Problem 4.66 in Kirby's List:
Problem 4.66 How do metrics (e.g. Riemannian, Lorentz, ...
- 1,897
0
votes
1
answer
526
views
How the distance between sets is called?
Hello,
I've recently write down some measure for sets and now I wonder how it is called or where it is described?
The measure itself is the following:
Let $A$ & $B$ -- two sets of values from a ...
- 103
42
votes
10
answers
5k
views
Is there a mathematical axiomatization of time (other than, perhaps, entropy)?
Since Euclid's axiomatization of space, we have developed a sophisticated mathematical model of space. Given a category of structures (measures), local space is modeled the spectrum of measurements ...
0
votes
1
answer
219
views
Find elements $\rho_i$ such that $H < B : [\langle H, \rho_i \rangle : H ] = 2$
Let $\Gamma (G;(G_i)_{i \in I})$ be a coset geometry (in the sense of Buekenhout) firm, residually connected and flag transitive with Borel subgroup $B$. Consider $H$ any subgroup of $B$. I want to ...
- 499
11
votes
2
answers
1k
views
Five points in spheres
Do there exist five points in the euclidean space ${\mathbb R}^3$ such that
every four of these points are in a spherical ball of radius 1, but that the five points are not in a ball of radius 1?
Do ...
- 111
2
votes
1
answer
370
views
Large subgroups of the Hamming cube
Let's consider the abelian group $\mathbb{Z}^N_2$ equipped with the Hamming metric (the hypercube).
Suppose I have a subgroup of this hypercube (not necessarily a subcube) which is generated by a set ...
2
votes
0
answers
409
views
Important lines in triangle - reverse problem
It is known that if three numbers $x,y,z$ are the lengths of the edges of some triangle, then there exists a triangle with medians of length $x,y,z$. Also, if $x,y,z>0$ (no condition imposed) there ...
- 2,222
13
votes
8
answers
3k
views
Applications of the notion of of Gromov-Hausdorff distance
I am looking for applications of the notion of Gromov-Hausdorff convergence to prove theorems that a priori have nothing to do with it. Examples that I am aware of (thanks to wikipedia and google):
...
9
votes
5
answers
13k
views
Get a point inside a polygon
I have a 2D polygon of arbitrary geometry. I need to find any point that is inside of that polygon. Taking the center won't work, because the polygon might not be convex. Is there a way to quickly ...
- 201
11
votes
3
answers
3k
views
polyhedra with equilateral pentagons faces
In page http://loki3.com/poly/isohedra.html around six polyhedra with equilateral pentagons as faces are shown: a pyritohedron, icositetrahedrons... Is there a complete list of this kind of polyhedra? ...
- 111
8
votes
2
answers
2k
views
Quasi-isometries vs Cayley Graphs
The following questions might be trivial, however, I couldn't solve them:
Let $G$ be generated by a finite symmetric set $S$. Suppose that $\Gamma(G,S)$ is the corresponding right Cayley graph of $G$...
- 244
18
votes
2
answers
1k
views
Example of a compact homogeneous metric space which is not a manifold
A metric space $(X,d)$ is isometrically homogeneous if its isometry group acts transitively on points, i.e., for every $x,y \in X$ there is an isometry $\varphi:X\to X$ with $\varphi(x) = y$. I'd ...
- 11.4k
9
votes
1
answer
523
views
The volume of the “unit ball” in $\mathbb{R}^{m\times n}$ with respect to the cut norm
This question is inspired by the question “ε-nets with respect to the cut norm” by the user Aaron, which had been reposted to cstheory.stackexchange.com.
The cut norm ||A||C of a matrix A=(aij)∈ℝm×n ...
- 1,959
1
vote
1
answer
342
views
Half-space comparison of perimeter
Claim: suppose that $E$ is a set of finite perimeter, and $H$ is a half space. Then $P(F\cap H)\le P(F)$. In words: restricting a Caccioppoli set to a half-space will not increase the perimeter.
My ...
- 320
5
votes
2
answers
1k
views
Geometric interpretation of $BN$-pairs
My question is relative to a geometric interpretation of the $BN$-pairs that arise in Tits' theory of buildings. Here is a definition that comes from an article by G. Stroth (Nonspherical spheres).
$[...
- 499
13
votes
3
answers
459
views
A comparison question for non-positively curved disks
Let $A$ and $B$ be two closed, 2-dimensional, non-positively-curved Riemannian disks (not necessarily with convex boundary). Suppose that their boundaries $\partial A$ and $\partial B$ have the same ...
- 56.6k
13
votes
1
answer
329
views
Spectral properties of finite metric sets
Given a finite metric set $S=\{P_1,\dots,P_n\}$, one gets a real symmetric matrix $M=M(S)$
with rows and columns indexed by elements of $S$ by setting
$M_{i,j}=d(P_i,P_j)$.
It is easy to see that $M$...
- 17.9k
15
votes
3
answers
1k
views
Stronger version of the isoperimetric inequality
I have been searching for a version of the isoperimetric inequality which is something like:
$P(\Omega) - 2\sqrt{\pi} Vol(\Omega)^{1/2} \geq \pi (r_{out}^2 - r_{in}^2)$ where $r_{out}$ and $r_{in}$ ...
- 2,641
10
votes
2
answers
930
views
What is determined by the combinatorics of the shadows of a convex polyhedron?
Define the shadow of a convex polyhedron $P$ in direction $u$
to be the orthogonal projection of $P$ onto a plane whose normal is $u$.
The shadow is a convex $k$-gon.
I am wondering to what degree $P$ ...
- 151k
8
votes
1
answer
1k
views
Is there an elementary way to show the triangular inequality for this expression ?
Consider the space $X$ of all scalar products on $\mathbb{R}^n$. For a scalar product $s$ and a base $B:=b_1\ldots,b_n$ let $M_{s,B}$ denote the matrix, whose $(i,j)$-th entry is $(s(b_i,b_j))$ . ...
- 11.1k
38
votes
10
answers
6k
views
Why is the Laplacian ubiquitous?
The title says it all.
I'm wondering why the Laplacian appears everywhere, e.g. number theory, Riemannian geometry, quantum mechanics, and representation theory. And people seems to care about their ...
- 3,806
3
votes
1
answer
386
views
Pointers/Papers on subdivision of planar quadrilateral meshes (PQ-Mesh) in 3D?
I'm interested in the subdivision of planar quadrilateral meshes (PQ-Meshes). Meshes consisting only of planar quadrilaterals, like discrete Voss surfaces and alike. I've been searching the web
for ...
- 133
6
votes
2
answers
656
views
Minimal surface which divides a convex body into two regions of equal volume
Question. Given a convex body $\Omega$, what is the shape of a surface $\Gamma$ of minimal area which divides $\Omega$ into two regions of equal volume?
Background/motivation.
A 2D version of the ...
- 22.3k
21
votes
6
answers
3k
views
Are there smooth bodies of constant width?
The standard Reuleaux triangle is not smooth, but the three
points of tangential discontinuity can be smoothed as
in the figure below (left), from the Wikipedia article.
However, it is unclear (to me) ...
- 151k
8
votes
0
answers
358
views
Coloring toroidal polyhedra with convex faces?
Consider a toroidal polyhedron, which is a topological torus, in which all faces are planar, two faces meet in at most an edge, and adjacent faces are not coplanar. The Szilassi polyhedron has 7 non-...
25
votes
3
answers
2k
views
Angle of a regular simplex
I find the following question embarrassing, but I have not been able to either resolve it, or to find a reference.
What is the vertex angle of a regular $n$-simplex?
Background: For a vertex $v$ ...
- 7,836
58
votes
14
answers
19k
views
Open problems in Euclidean geometry?
What are some (research level) open problems in Euclidean geometry ?
(Edit: I ask just out of curiosity, to understand how -and if- nowadays this is not a "dead" field yet)
I should clarify a bit ...
7
votes
3
answers
792
views
Shadow boundary on convex body in $\mathbb{R}^3$
Let $S$ be the surface of a compact, convex, smooth ($C^\infty$) body in $\mathbb{R}^3$,
with strictly positive Gaussian curvature at every point of $S$.
Fix a direction $z$ in a Cartesian coordinate ...
- 151k
18
votes
2
answers
2k
views
Which platonic solids can form a topological torus?
8 cubes can be joined face-to-face to form a closed ring with a hole in it, with each cube sharing a face with only two others. The same can be done with 8 dodecahedrons.
Is the same possible with the ...
- 453
13
votes
0
answers
751
views
$\epsilon$-nets with respect to the cut norm
The cut norm $||A||\_C$ of a real matrix $A = (a_{i,j}) \in \mathcal{R}^{n\times n}$ is the maximum over all $I \subseteq [n], J \subseteq [n]$ of the quantity $\left|\sum_{i \in I, j \in J}a_{i,j}\...
- 794
17
votes
3
answers
2k
views
Decidability of tiling R^2
Does there exist a closed curve, with finite area and finite circumference, of which it is undecidable (in an axiomatic system where it is constructable) whether it can tile the plane?
I know the ...
- 453
5
votes
1
answer
395
views
Average distance to a curve of fixed length
Let $C$ be a continuous curve in the unit square having length $L$. Is there a lower bound on the average distance between the points in the unit square and $C$, as a function of $L$? Is there an ...
- 1,125
3
votes
0
answers
233
views
How many set partitions on a big cube’s boundary arise from cubomino decompositions of the solid cube?
Introduction. This is a counting question about configurations that can appear on the outside of assembled Soma cube-like puzzles. More specifically, it’s about the ways in which the pieces of an ...
- 588