All Questions
4,827 questions
15
votes
1
answer
699
views
Continuity in terms of lines
Let $f: \mathbb R^n \rightarrow \mathbb R^n$, where $n> 1$ be a bijective map such that the image of every line is a line.
Is $f$ continuous?
I think it is, but the proof isn't immediately ...
- 230
14
votes
1
answer
886
views
Distance to an apartment of the affine building of GL(N)
Here $F$ is a locally compact non-archimedean non-discrete field.
Let $X$ be the reduced (affine) Bruhat-Tits building of ${\rm GL}(n,F)$. Fix a maximal split torus $T$. Let $B$ be a Borel subgroup ...
- 6,349
63
votes
8
answers
14k
views
Fair but irregular polyhedral dice
I am interested in determining a collection of geometric conditions that will guarantee that a convex polyhedron
of $n$ faces is a fair die in the sense that, upon random rolling, it has an equal $1/n$...
- 151k
5
votes
2
answers
565
views
Zorn's Lemma and plane geometry
Given a graph $G$ and a number $n$, Zorn's Lemma immediately implies the existence of a
maximal partial coloring of $G$. Equivalently, one may assign $n+1$ colors to the nodes of $G$ such that nodes ...
- 17.6k
11
votes
2
answers
2k
views
Non-Kahler Complex manifolds
For a non-Kahler complex manifold $M$, we still have the decomposition of differential forms into differential forms of type $(p,q)$ and we can write $d=\partial+\bar\partial$ and we can define ...
22
votes
5
answers
3k
views
Unexpected applications of Dvoretzky's theorem
Dvoretzky's theorem is a classic of convex geometry. Recently at a conference in quantum information I learned (from Patrick Hayden's talk) about a nontrivial application of the theorem to a problem ...
- 2,449
16
votes
1
answer
511
views
Subdividing a polyhedral space into convex simplices
A (Euclidean) polyhedral space is a metric space obtained by "gluing together" several (let's assume finitely many) Euclidean simplices (of varying dimensions) by identifying some faces via isometries....
- 32.4k
6
votes
2
answers
407
views
Regularity of asymptotic cones
Are there any general conditions guaranteeing that the asymptotic cone of a group/graph is "regular" in some sense? E.g. for $\mathbb{Z}^d$ we get $\mathbb{R}^d$ as the asymptotic cone, which is even ...
- 3,323
7
votes
1
answer
456
views
Space-discriminating injective curve
Let $f\colon \mathbb R^1\to \mathbb R^3$ be a continuous and injective map. Is $\mathbb R^3\setminus f(\mathbb R^1)$ a path-connected space?
- 5,063
18
votes
3
answers
2k
views
Are the Platonic solids shadows of 4-polytopes?
Say that a 3D shadow of a 4-polytope is a parallel projection to 3-space, not necessarily orthogonal to that 3-space (that would make it an orthogonal projection).
I am wondering if each of the five ...
- 151k
6
votes
2
answers
643
views
English translation of Lambert's Theorie der Parallellinien?
Does anyone know if there is an available (published or unpublished) English translation of Johann Lambert's Theorie der Parallellinien? I was able to find it online in German by way of the ...
- 533
2
votes
0
answers
160
views
Tubular neighborhood growth of zero set of polynomial of bounded degree in the torus
This question is related to my related post:
Volume growth of tubular neigbhorhood of critical values of an algebraic/differentiable map
The setting here is as follows:
Let $p: \mathbb{R}^{2k} \to \...
- 4,466
15
votes
2
answers
2k
views
Partitioning a Rectangle into Congruent Isosceles Triangles
Is it possible to partition any rectangle into congruent isosceles triangles?
- 404
9
votes
1
answer
1k
views
Rigidity of triangle comparison in Alexandrov spaces
For $CAT(\kappa)$ spaces $X$ we have following rigidity result: if equality holds in any of the comparison distances between a triangle $\Delta$ in $X$ and the corresponding comparison triangle $\...
- 265
17
votes
1
answer
1k
views
Essentially one random metric on $\mathbb{S}^2$?
I heard it claimed that there is, in some sense, only
one random metric on $\mathbb{S}^2$.
I would appreciate any pointer to literature that explicates
this intriguing claim.
So far my own searches ...
- 151k
14
votes
0
answers
4k
views
Minimum tiling of a rectangle by squares
Given the $n\times m$ rectangle, I want to compute the minimum number of integer-sided squares needed to tile it (possibly of different sizes).
Is there an efficient way to calculate this?
- 1,015
33
votes
4
answers
3k
views
Does there exist a shot in ideal pocket billiards?
Assume you have one shot with the cue ball in pocket billiards (a.k.a. pool), with
the game idealized in that no spin is placed on the cue ball in
the initial shot, all collisions between billiard ...
- 151k
0
votes
2
answers
1k
views
What is the average center of six points in space
I have three pairs of points in 3D space. These may or may not be coplanar. I want to find a point such that it is equidistant from each pair of points. I know that may or may not be possible ...
- 25
17
votes
3
answers
3k
views
Nonseparable example in dimension theory?
Could you give me an example of a complete metric space with covering dimension $> n$ all of which closed separable subsets have covering dimension $\le n$?
The question closely related to this ...
- 1,785
25
votes
1
answer
7k
views
Hanging a ball with string
What is the shortest length of string that suffices to hang
a unit-radius ball $B$?
This question is related to an earlier MO question, but I think different.
Assume that the ball is frictionless.
...
- 151k
4
votes
1
answer
778
views
Example in dimension theory
Could you give me an example of a complete metric space wiht covering dimension $> n$ all of which compact subsets have covering dimension $\le n$?
- 1,785
11
votes
2
answers
3k
views
Levy's isoperimetric inequality for sphere
Let me recall subj:
If $s>0$, $A$ and $B$ are two subsets of $\mathbb{S}^{n}$, $|A|=|B|$ ($|\cdot|$ stands for the Lebesgue measure on the sphere) and $B$ is a cup $B=\{ (x_1,x_2,\dots,x_n)\in \...
- 109k
9
votes
3
answers
2k
views
Approximation by locally Lipschitz functions
Could you tell me what is the name and/or reference for the following theorem:
Let $M$ be a metric space. Then any continuous function $f:M\to\mathbb R$ can be a be uniformly approximated by a ...
- 1,785
11
votes
6
answers
1k
views
Decomposing the plane into intervals
I posted this on Stack Exchange and got a lot of interest, but no answer.
A recent Missouri State problem stated that it is easy to decompose the plane into half-open intervals and asked us to do so ...
- 356
10
votes
2
answers
2k
views
Is there a compass and straightedge construction of parallel lines in hyperbolic geometry?
Is there a compass and straightedge construction of parallel lines in hyperbolic geometry?
That is, given a line and a point not on the line, construct a line parallel to the given line.
- 284
2
votes
2
answers
2k
views
Anuloid (Torus) x line intersection
Hi,
I need calculate ray (line) intersection with torus for my ray-tracing program (I know, its to graphics, but i need math behind it).
I can solve equation of order x^4, but thats too way slow (...
- 21
1
vote
1
answer
390
views
Isocontours of depth and magnitude of gradient
We are interested in characterizing a 2D surface $z(x,y)$, where $(x,y)$ is the regular 2D Cartesian grid. Let $\nabla z = (z_x, z_y)$ denote the gradient. The surface is a "general" one, that is, ...
- 13
12
votes
1
answer
1k
views
Cobounded ⇒ cocompact?
Assume $\Gamma$ acts by isometries on a separable Hilbert space $H$, and
$$\operatorname{diam} H/\Gamma\le 1.$$
Is it true that $H/\Gamma$ is compact?
Stupid example. Assume the action of $\...
- 45k
2
votes
1
answer
1k
views
Linear Programming Cost Function [closed]
I need to add the following to my LP problem:
If the amount of workers hired in period $t$ ($H_t$) is higher than 25, the hiring cost is only 1 instead of 1.2.
Example: if 30 workers are hired in ...
5
votes
2
answers
459
views
How indepenedent of a chosen metric is the box-counting dimension? Is there a non-integral dimension which is defined for topological spaces?
Question 1. Given a topological space $X$ and two metrics $a$ and $b$ on it, compatible with the topology, what conditions should I impose on them so that box-counting (or other, for example Hausdorff)...
- 3,897
8
votes
1
answer
199
views
minimal diameter of full preimage of torus
Given a set $A\subset \mathbb{R}^n$ such that $A\cap (x+\mathbb{Z}^n)\ne \emptyset$ for any $x\in \mathbb{R}^n$ (that is, $p(A)=\mathbb{T}^n$ for the projection $p:\mathbb{R}^n\rightarrow \mathbb{T}^...
- 109k
6
votes
3
answers
913
views
Metric spaces as algebraic systems
Let $(X, {\mathrm{dist}})$ be a metric space. In the paper by Kramer, Shelah, Tent and Thomas , they define an algebraic system $A(X)$ as the set $X$ with countably many binary relations $R_\alpha$, ...
user6976
2
votes
1
answer
743
views
weak metric space
In the definition of a metric space, replace the triangle inequality by the weaker inequality
d (x, z) ≤ C max {d (x, y), d (y, z)},
where C is a positive constant (depending on the "metric", ...
11
votes
2
answers
2k
views
Midpoint geodesic polygon / Birkhoff curve shortening
I would like to know under what conditions the process
of creating a midpoint piecewise geodesic polygon converges
on a surface $S \subset \mathbb{R}^3$.
$S$ may be assumed smooth, closed, and ...
- 151k
1
vote
4
answers
978
views
Maximum average value within a rectangular bounding box
The goal is to expedite detection using the sliding window approach. In other words, an object classifier is known and I need to find where the possible locations of this object are in an image. This ...
- 111
8
votes
2
answers
484
views
Preferred embedding of finite metric spaces in riemaniann manifolds of given dimension
In search for a Machian formulation of mechanics I find the following problem. In Machian mechanics absolute space does not exists, and the only real entities are the relative distances between the ...
- 339
9
votes
4
answers
631
views
Estimating direction from a distribution on a circle
Let there be $n$ points on a unit circle. It is known they come from "normal" distribution around particular unknown direction (i.e. sum of 2 "normal" distributions on circle - one centered at point $...
- 93
1
vote
2
answers
1k
views
Inequality-constrained linear-regression, what is the covariance of the estimator?
If you do a linear regression: $||Ax - e ||^2$, where e is iid Gaussian, mean 0 and variance 1, then your answer is $x_{hat} = (A' A)^{-1} (A' * e)$ and the covariance of $x_{hat}$ is $(A' A)^{-1}$
...
6
votes
1
answer
767
views
Using mirrors to make a non-convex polygon visible from a fixed interior point
Take a point $A$ inside a non-convex polygon $P$. Is it always possible to place a finite set of mirrors given by straight segments (not necessarily along the boundary of $P$, any position inside $P$ ...
- 17.9k
5
votes
3
answers
2k
views
Optimal packing of spheres tangent to a central sphere
Please consider a central, ordinary 2-sphere $S_1$, of some radius $r_1$, and a second ordinary sphere, $S_2$, of radius $r_2$, where $r_2 \leq r_1$.
My question concerns optimal values for the ...
13
votes
3
answers
2k
views
Metric angles in Riemannian manifolds of low regularity
Given three points $a,b,c$ in a (geodesic) metric space $X$, one defines a comparison angle $\angle(a,b,c)$ by the cosine law:
$$
\angle(a,b,c) = \arccos \frac{|ab|^2 + |ac|^2 - |bc|^2}{2\cdot|ab|\...
- 32.4k
5
votes
1
answer
569
views
Maximum distance to nearest-lattice-point on (hyper-)sphere with unit lat-lon lattice.
Let $U$ be the set of all non-null $n \times 1$ vectors $\mathbf{\mathrm{u}}$, where $u_i \in \lbrace-1, 0, 1\rbrace$. Let $\mathbf{\mathrm{x}}$ be an $n \times 1$ vector in $\mathbf{R}^n$. Let $\...
6
votes
0
answers
2k
views
Geometric Proof that Fubini-Study Metric is Round
The Fubini-Study metric d(x,y) on $CP^1$ is defined as follows: for x and y in $CP^1$ let v and w be unit vectors in $C^2$ representing x and y. Then $d(x,y)=2arccos(\langle v,w\rangle)$. The round ...
- 159
11
votes
2
answers
489
views
Shortest morphing between shapes embedded in $\mathbb{R}^3$
I am interested in what in computer graphics is called
morphing between two topologically equivalent shapes $S_0$
and $S_1$ in 3D.
This is a continuous "path" of shapes $S_t$, each embedded and
all ...
- 151k
1
vote
1
answer
271
views
Synthetic Proof for Ratio of Volumes of Concentric Spheres?
Let $B^n(r)$ be the $n$-ball of radius $r$. A standard (easy) problem for first year calculus students is the following.
$(1)$ Show that $$ \lim_{n\to \infty} \frac{\text{Vol}(B^n(r))}{\text{Vol}...
- 23k
10
votes
2
answers
4k
views
Morphism between projective varieties
Let $f:X \rightarrow Y$ be a morphism between two smooth projective varieties $X,Y$ which are defined over an algebraically closed field $k$. I am looking for some criteria which guaranties the ...
- 690
3
votes
1
answer
270
views
When is a blow-up a non-trivial product?
Suppose $X$ is an algebraic variety and let $Z \subset X$ be a subvariety. Are there some useful criteria under which the blow-up $Bl_Z X$ becomes a nontrivial product $V \times W$ of the algebraic ...
- 690
26
votes
6
answers
3k
views
Easy proof of the fact that isotropic spaces are Euclidean
Let $X$ be a finite-dimensional Banach space whose isometry group acts transitively on the set of lines (or, equivalently, on the unit sphere: for every two unit-norm vectors $x,y\in X$ there exist a ...
- 32.4k
6
votes
2
answers
945
views
Vortex Voronoi diagram?
Suppose there are a finite number of disjoint unit-radii disks in the
plane, each spinning clockwise or counterclockwise at the same
angular velocity.
The plane is filled with a thin fluid layer,
and ...
- 151k
10
votes
1
answer
560
views
Are packing-homogeneous spaces homogeneous?
Given a metric space (M,d) define the packing function P(x,R,r) to be the maximum number of non-intersecting balls of radius r with centers in the ball B(x,R). Let’s call M packing-homogeneous if the ...
