All Questions
4,827 questions
11
votes
2
answers
587
views
Algebraicity of the completion of a field? Finiteness?
At the end of my 8410 class today (see http://alpha.math.uga.edu/~pete/MATH8410.html if you care), one of my students asked me the following very interesting question:
Let $(K,|\ |)$ be a normed field,...
- 65.4k
6
votes
1
answer
802
views
Approximation of a Sobolev function that has vanishing trace on the reduced boundary of a Caccioppoli (i.e. finite perimeter) set
For $\Omega\subset\mathbb{R}^N$ open and bounded, let $W^{1,p}(\Omega)$ denote the usual Sobolev space of $L^p(\Omega)$ functions with weak partial derivatives in $L^p(\Omega)$ and $W_0^{1,p}(\Omega)$ ...
- 143
2
votes
3
answers
490
views
find the collision of a particle with a swept triangle.
Given there is triangle: V in 3D space that transforms over time t -> t1 to V1, and a static point P is somewhere in 3d space, how can I determine if P ever collides with V, and if so at what value of ...
- 123
4
votes
3
answers
554
views
Uniqueness of a polygon
Suppose I have two $n$-sided polygons A and B. Is there a non-trivial upper bound on the number of parameters (eg. area, perimeter, etc) of the two polygons, that need to be the same, for A and B to ...
- 75
28
votes
8
answers
6k
views
Representability of finite metric spaces
There have been a couple questions recently regarding metric spaces, which got me thinking a bit about representation theorems for finite metric spaces.
Suppose $X$ is a set equipped with a metric $d$...
- 4,014
14
votes
2
answers
878
views
What is the prime spectrum of a Cauchy series ring?
Let $k$ be a field, and let $| \ |$ be a norm on $k$. The norm induces a metric. To construct the completion $\hat{k}$ as a normed field, the standard recipe is to take the quotient of the ring $\...
- 65.4k
9
votes
3
answers
944
views
Can different bicycles leave the same tracks?
(asked by JST on the Q&A board at JMM)
Can two bicycles of different lengths leave the same set of tracks (aside from a straight line)?
- 1,119
6
votes
2
answers
365
views
Why is GL(n,C)/U(n) a CAT(0) space?
The title says it all. In one of his answers to the question "Convex hull in CAT(0)" (I don't have the points to post a link, if someone doesn't mind link-ifying this that would be cool), Greg ...
- 2,869
26
votes
4
answers
4k
views
What is the "right" universal property of the completion of a metric space?
I'm a little embarrassed to ask this one, but it could help for a class I'm teaching, so here goes:
Let $X$ be a metric space. We all know that $X$ admits a completion, which is a complete metric ...
- 65.4k
6
votes
2
answers
6k
views
Minimum-area bounding quadrilateral algorithm
There are a few algorithms around for finding the minimal bounding rectangle (OBB) containing a given (convex) polygon.
Does anybody know about an algorithm for finding a minimal-area bounding ...
- 163
18
votes
3
answers
3k
views
Deciding membership in a convex hull
Given points $u, v_1, \dots,v_n \in \mathbb{R}^m$, decide if $u$ is contained in the convex hull of $v_1, \dots, v_n$.
This can be done efficiently by linear programming (time polynomial in $n,m$) in ...
- 667
6
votes
3
answers
1k
views
How can I embed an N-points metric space to a hypercube with low distortion?
I have a N-point metric space defined by the pairwise distance matrix. I want to encode these N points with binary strings, i.e. each point will be mapped to a vertex in a hypercube. The lengths of ...
- 171
1
vote
1
answer
419
views
Is the direction of the longest line of a polytope unique?
The question pertains to a polytope that is generated by the intersection of an affine subspace with a hypercube in $p$ dimensions.
The affine subspace is given by:
$X \mbox{ u} = y$
where
$u$ &...
- 121
9
votes
2
answers
560
views
Integrating a simple exponential over the space of matrices that define a metric
I want to interpret an $n\times n$ matrix $D$ as a set of pairwise distances, and assume that $D$ obeys metric properties. Namely, $D_{ii} = 0$, $D_{ij} \geq 0$, $D_{ij} = D_{ji}$ and $D_{ij} \leq D_{...
- 147
4
votes
3
answers
852
views
Variational characterization of curvature?
Consider a surface $S$ smoothly embedded in $\mathbb{R}^3$. Classically, the (Riemannian) curvature of $S$ is described by the second fundamental form, which is constructed from partial derivatives ...
- 3,059
7
votes
2
answers
366
views
Simplicial and cubical decompositions of low valence
Every surface can be triangulated in such a way that at most 7 trianlges meet at one vertex. Every surface can be decomposed in squares such that at every vertex at most 5 suqares meet. For surfaces ...
- 28.9k
4
votes
2
answers
818
views
Number of independent distances between n points in d-dimensional Euclidean space?
There are $\binom{n}{2}$ distances between $n$ points in $\mathbb{R}^d$. Not all of them can be chosen freely if $n$ exceeds the number $n_d = d + 1$. If $n = n_d$ we obviously have $\binom{d+1}{2}$ ...
- 9,720
7
votes
5
answers
1k
views
How to compute the average distance till intersection within a triangle in $\mathbb{R}^2$?
You are given 3 points in $\mathbb{R}^2$; $A$, $B$, $C$ forming a triangle with area > 0. You pick an arbitrary point inside $ABC$ and an arbitrary direction. After some distance $d$, you will ...
- 171
28
votes
7
answers
5k
views
Rolle's theorem in n dimensions
This looks like a statement from a calculus textbook, which perhaps it should be.
"Rolle's theorem". Let $F\colon [a,b]\to\mathbb R^n$ be a continuous function such that $F(a)=F(b)$ and $F'(t)$ ...
7
votes
2
answers
726
views
Zeta function for curves in a manifold
Motivation
In the analogy between prime numbers and knots, the prime number is thought sometimes as the circle of length $l([p]) = \text{log}\,p$. This is so you can express the zeta function as
$$ \...
- 15.1k
1
vote
2
answers
6k
views
If a quadratic form is positive definite on a convex set, is it convex on that set?
Consider a real symmetric matrix $A\in\mathbb{R}^{n \times n}$. The associated quadratic form $x^T A x$ is a convex function on all of $\mathbb{R}^n$ iff $A$ is positive semidefinite, i.e., if $x^T A ...
- 3,059
3
votes
2
answers
2k
views
Cone angles for Riemannian metrics in polar coordinates
This is the simplest case of a question that's been bugging me for a while: say we have a Riemannian metric in polar coordinates on a $(2-d)$ surface:
$$
g=dr^2+f^2(r, \theta;)d\theta^2,
$$ such that ...
- 1,665
-4
votes
1
answer
8k
views
How to transform a plane into a sphere? [SOLVED] [closed]
Given a 2-dimensional array of MxN heights, how to transform it to a sphere? Every element of this array is just a 3D point (x,y,z) where z represents some height. One has to transform this array into ...
- 555
12
votes
3
answers
1k
views
distance regular metric spaces
A metric space (V,d) will be called distance regular if for every distances a>0, b, c a nonnegative integer p(a,b,c) is defined, so that whenever d(B,C)=a, there are precisely p(a,b,c) points A ...
26
votes
7
answers
10k
views
Uniformly Sampling from Convex Polytopes
How to choose a point uniformly from a convex polytope $P \subset [0,1]^n$ defined by some inequalities, $Ax < b$? (Here $A$ is an $m \times n$ matrix, $x \in \mathbb{R}^n$, and $b \in \mathbb{R}^...
- 22.8k
-4
votes
4
answers
677
views
What is the max number of points in R^3, interconnected by generic curves?
The largest complete graph that embeds in 2 dimensions is $K_4$, while the largest complete graph that embeds in 3 dimensions is $K_{\infty}$, right? However, I don't know any constructive proof of it....
- 555
8
votes
6
answers
1k
views
Combinatorial distance ≡ Euclidean distance
Definition: A polytope has property X iff there is a function f:N+ → R+ such that for each pair of vertices vi, vj the following holds:
disteuclidean(vi, vj) = f(distcombinatorial(vi, vj))
with ...
- 9,720
7
votes
1
answer
2k
views
Why is 3 a bad constant in the Vitali covering lemma?
Hi,
recently I had to do with the Hardy-Littlewood maximal function and we used there the Vitali covering lemma with constant 5. Then, given an advice, I proved it with constant k>3. But I cannot ...
- 93
11
votes
4
answers
958
views
Geometry of the multilagrangian Grassmannian
Let's introduce the following variety $MG(3,6)$, which is a "multisymplectic" analog of a Lagrangian Grassmannian $LG(3,6)$.
Consider a 3-form $\omega = dx1 \wedge dx2 \wedge dx^3 - dx4 \wedge dx5 \...
- 1,783
10
votes
1
answer
835
views
what was Hilbert's geometric construction in his 17th problem?
Hilbert's 17th problem asked if a nonnegative real polynomial is the sum of squares of rational functions. It was answered affirmative by Artin in around 1920. However, in his speech, he also asked if ...
user2529
15
votes
5
answers
1k
views
Diameter of universal cover
Let $M$ be Riemannian manifold and $\tilde M$ be its universal cover (with induced metric).
What is the upper bound for $k=\mathop{diam}\tilde M/\mathop{diam} M$ in terms of $m=|\pi_1(M)|$ (or $\pi_1(...
11
votes
2
answers
2k
views
Characterization of Riemannian metrics
This is probably an insanely hard question, but given an abstract metric space, is there some way to determine whether it's a manifold with a Riemannian, or more generally a Finslerian, metric? If ...
- 1,665
7
votes
2
answers
1k
views
Example of non-closed convex hull in a CAT(0) space
this is related to this question but is simpler, and hopefully is well-known. There are a number of references that say that the convex hull of a collection of points in a CAT(0) space need not be ...
- 4,462
3
votes
1
answer
439
views
Convex n-polytope general position vectors to general position vectors of tetrahedron
I asked this question in a comment to this question, but got no response. I thought that perhaps it needed more exposure, so I made it a question in itself.
Define a set of general position vectors $...
- 4,842
124
votes
37
answers
12k
views
One-step problems in geometry
I'm collecting advanced exercises in geometry. Ideally, each exercise should be solved by one trick and this trick should be useful elsewhere (say it gives an essential idea in some theory).
If you ...
19
votes
2
answers
1k
views
Four Dimensional Origami Axioms
What are the axioms of four dimensional Origami.
If standard Origami is considered three dimensional, it has points, lines, surfaces and folds to create a three dimensional form from the folded ...
- 193
1
vote
1
answer
335
views
Systems of conics
It seems well-known that the system of conics given by $\frac{x^2}{a^2}+\frac{y^2}{a^2-c^2}=1$ for $c>0$ fixed and $a \in (0,c)\cup(c,\infty)$ varying is orthogonal: whenever two of these curves ...
- 13.1k
97
votes
11
answers
13k
views
Is it possible to capture a sphere in a knot?
You and I decide to play a game:
To start off with, I provide you with a frictionless, perfectly spherical sphere, along with a frictionless, unstretchable, infinitely thin magical rope. This rope ...
- 8,688
9
votes
5
answers
2k
views
Feasibility of a list of prescribed distances in R^3
I am puzzled with the following problem:
Given $n$ real numbers it is to obtain a Yes/No answer to: "whether it is possible to arrange different points in the Euclidean $\mathbb{R}^3$ so that every ...
- 555
30
votes
3
answers
1k
views
Diameter of m-fold cover
Let $M$ be a closed Riemannian manifold.
Assume $\tilde M$ is a connected Riemannian $m$-fold cover of $M$.
Is it true that
$$\mathop{diam}\tilde M\le m\cdot \mathop{diam} M\ ?\ \ \ \ \ \ \ (*)$$
...
4
votes
2
answers
3k
views
Break polyhedron into tetrahedron
Given a polyhedron consists of a list of vertices (v), a list of edges (e), and a list of surfaces connecting those edges (...
- 381
23
votes
2
answers
2k
views
Is there a neat formula for the volume of a tetrahedron on $S^3$?
There is a nice formula for the area of a triangle on the 2-dimensional sphere;
If the triangle is the intersection of three half spheres, and has angles $\alpha$, $\beta$ and $\gamma$, and we ...
- 435
17
votes
1
answer
1k
views
Tropical mathematics and enriched category theory
Is there a connection between tropical mathematics and the Lawvere enriched category theory approach to metric spaces? I guess I will give a partial answer to this below, but I mean can they be ...
- 1,271
27
votes
6
answers
2k
views
When shorter means smaller?
Assume a convex figure $F\subset \mathbb R^2$ satisfies the following property: if $f:F\to \mathbb R^2$ is a distance-non-increasing map then its image $f(F)$ is congruent to a subset of $F$.
Is it ...
18
votes
1
answer
644
views
Egalitarian measures
A question I got asked I while ago:
If $T$ is a triangle in $\mathbb R^2$, is there a function $f:T\to\mathbb R$ such that the integral of $f$ over each straight segment connecting two points in the ...
- 47.3k
7
votes
2
answers
846
views
What is known about polyhedra nets that allow overlapping?
It is an open problem that the net of any convex polyhedron can be unfolded onto a flat plane with no overlapping. Is anything known if we allow x faces to overlap? For example, is it known if any ...
- 2,615
28
votes
8
answers
5k
views
Convex hull in CAT(0)
Let $X$ be complete $\mathop{CAT}(0)$-space and $K\subset X$ be a compact subset.
Is it true that convex hull of $K$ is compact?
Comments:
Convex hull of $K$ = intersection of all closed convex sets ...
- 45k
9
votes
1
answer
604
views
Which changes of metric fix all open balls of a metric space?
In an earlier question, I was interested in counting the number of metric spaces on N points, where I considered two metric spaces to be the same if they had the same collection of open balls. Two ...
- 1,861
14
votes
5
answers
5k
views
Explanation for gamma function in formula for $n$-ball volume
It is well-known that the volume of the unit ball in n-space is $\pi^{n/2}/\Gamma(n/2+1)$. Do you know of a proof which explains this formula? Any proof which does not treat the cases $n$ even and $n$ ...
- 141
11
votes
7
answers
1k
views
What are some interesting ways of making new metrics out of old metrics?
If $d(x,y)$ and $e(x,y)$ are metrics then $d(x,y)+e(x,y)$ and $\frac{d(x,y)}{1+d(x,y)}$ are metrics.
If $d_i(x,y)$ for $i=1,\dots,n$ are metrics then so is $\sqrt{\sum_{i=1}^n{d_i^2(x,y)}}$
Are ...
- 3,613