Tagged Questions
Euclidean, hyperbolic, discrete, convex, coarse geometry, comparisons in Riemannian geometry, symmetric spaces.
0
votes
0answers
2 views
Are $\varepsilon$-connected components dense?
Let $X$ be a connected compact metric space. Given a positive $\varepsilon$ and two points $x,y\in X$ we write $x\sim_\varepsilon y$ if there exists a sequence $C_1,\dots,C_n$ of connected subsets of ...
8
votes
0answers
53 views
Is every metric continuum almost path-connected?
The question was motivated by this question of Anton Petrunin.
By a metric continuum we understand a connected compact metric space.
Let $p$ be a positive real number. A metric continuum $X$ is ...
5
votes
1answer
73 views
Contractibility of balls in Alexandrov spaces
Let $X$ be a compact finite dimensional Alexandrov space with curvature bounded below.
Does there exist $\varepsilon_0>0$ (depending on $X$) such that for any $\varepsilon \in (0,\varepsilon_0)$...
5
votes
0answers
95 views
Balls in Lawvere metric spaces
Let $V$ be the monoidal category $[0,\infty)$ (as a poset) with $+$ and $0$. Lawvere shows that $V$-enriched categories are a more natural generalisation of the notion of a metric space (note no ...
2
votes
1answer
167 views
Name and Algorithms for a Sparsest Circle Packing
The ordinary circle packing problem in the variant with equal radii asks for the largest radius $r_{max}$ that allows placing $n$ non-overlapping circles with radius $r_{max}$ e.g. in the unit square, ...
4
votes
1answer
72 views
KL divergence and mixture of Gaussians
Do we have an exact formula to compute the KL divergence between 2 mixtures of Gaussians (i.e convex combinations of a finite number of Gaussian distributions)?
If not exactly known, are there good ...
5
votes
0answers
88 views
How to pack 27 $a\times b\times c$ blocks into a cube of side $a+b+c$ with some kind of symmetry?
Recently I stumbled on the problem quoted here about a geometric proof of the AM-GM inequality $$(a_1+\cdots+a_n)^n\ge n^n a_1\cdots a_n$$ by packing $n^n$ rectangular $ n$-dimensional boxes of sides $...
25
votes
1answer
441 views
+500
Running most of the time in a connected set
Let $P$ be a compact connected set in the plane and $x,y\in P$.
Is it always possible to connect $x$ to $y$ by a path $\gamma$ such that the length of $\gamma\backslash P$ is arbitrary small?
...
8
votes
2answers
216 views
Is it possible to continuously select a probability distribution over fixed points in Brouwer's fixed point theorem?
According to Brouwer's fixed point theorem, for compact convex $K\subset\mathbb{R}^n$, every continuous map $K\rightarrow K$ has a fixed point.
However, these fixed points cannot be chosen ...
1
vote
1answer
84 views
Chain rotation of a point
Let $n$ be a positive integer number and $P$ be a point in a plane. Let $A_1$, $A_2$, $\cdots$, $A_m$ be $m$ points in the plane, we take modulo $m$ for $A_j$ (it is mean $A_{m+i}=A_{i}$ for $i=1, 2, \...
5
votes
1answer
139 views
Discrete approximations of Riemannian manifolds
MSE crosspost
It's known (due to Perelman) that in class of Alexandrov spaces of fixed dimension and bounded from below curvature Gromov-Hausdorff distance separates homeomorphism types — every $\...
0
votes
0answers
90 views
Integer reduction of a positive semidefinite matrix
Take $\Sigma $ a real positive semidefinite matrix. Define $P$ to be the smallest projection with the property that for any $\mathbf{a}\in \mathbb{Z}^n$ with $\mathbf{a}^\dagger (I-P)^\dagger \Sigma (...
6
votes
0answers
94 views
Tiling with incommensurate triangles
Say that two triangles are incommensurate if they do not
share an edge length or a vertex angle, and their areas differ.
Suppose you'd like to tile the plane with pairwise incommensurate triangles.
I ...
7
votes
2answers
194 views
Non-separable metric probability space
Let us say a metric probability space $(X,\rho,\mu)$ has property (*) if:
the support of $\mu$ is contained in a separable subspace of $X$.
Questions:
1. Is there a standard name for this property?
...
0
votes
0answers
24 views
Simultaneous movement toward barycenters - what can be guaranteed
Suppose a tiling is given in 2D (an embedding of a planar triangulated graph), with all faces convex.
Now suppose one moves each point, one by one, to the barycenter of its neighbors. I think that ...
3
votes
2answers
71 views
Maximizing minimal distance between consecutive brushstrokes when painting a checkerboard torus
Suppose you have a 2-torus and you want to paint an $m\times n$ checkerboard pattern on it.
Every brushstroke could paint a single square.
How does one maximize the minimal distance between ...
5
votes
1answer
185 views
Is the action of $SO(n)$ on the sphere $S^{n-1}$ ballanced?
A subset $B$ of a group $G$ is called balanced if $gBg^{-1}=B$ for all $g\in G$.
An action of a group $G$ on a metric space $X$ is called ballanced if for each non-empty balanced subset $B\subset G$ ...
0
votes
1answer
56 views
Hausdorff convergence of submanifolds in $\mathbb{S}^m$
Let $\{X_i^n\}_{i\in \mathbb{N}}$ and $\{Y_i^n\}_{i\in \mathbb{N}}$ be sequences of connected closed submanifolds of $\mathbb{S}^{n+2}$, with $n> 5$. Suppose that $\{X_i^n\}_{i\in \mathbb{N}}$ (...
-2
votes
1answer
426 views
Is the conjecture true for n-sphere $(n>2)$? [closed]
This is higher dimension conjecture of Problem 3845 in Crux Mathematicorum and Theorem 2 in here:
PS: This figure is very nice, this is also generalization of Brianchon’s theorem, The Pascal theorem, ...
0
votes
0answers
112 views
Expected distance in hyperbolic space
In a hyperbolic space, $r$ and $\theta$ can represent a point in a polar coordinate system. If we suppose $\theta_1\sim \operatorname{Uniform}(t_1,t_2)$, $\theta_2\sim \operatorname{Uniform}(t_3,t_4)$,...
3
votes
0answers
38 views
Measure of set of vectors whose outer product are bounded
Let consider the canonical Euclidean space $E = \mathbb{R}^n$, endowed with the Lebesgue measure $\mu$.
Define the map $v_k: E^k \rightarrow \mathbb{R}$ that
sends a $k$-uple $x_1,\cdots, x_k$ of ...
1
vote
0answers
35 views
From Sudakov minoration principle to lowerbounds on Rademacher complexity
For a compact subset $S \subset \mathbb{R}^n$ (and an implicit metric $d$ on it) and $\epsilon >0$ lets define the following $2$ standard quantities,
Let ${\cal P}(\epsilon,S,d)$ be the $\epsilon-...
3
votes
1answer
85 views
Smoothness of a curve vs. smoothness of the squared distance from the curve to points on Riemann manifolds
I know that the squared distance function from a point $p$ on a Riemann manifold $M$ is smooth in a n-hood of $p$. Therefore for a smooth curve $c:\mathbb{R}\to M$ the concatenation $d(p,\cdot)^2\circ ...
2
votes
1answer
188 views
Some inequalities on chain of circle packing
By my computation, I pose a conjecture as follows and I am looking for a proof:
Conjecture: Let $(O)$ be a circle with radius $R$, and $n$ be positive integer $n\ge 3$. Construct $n$ circles $(...
5
votes
3answers
159 views
Average caliper diameter (mean width) of a polyhedron
Define the caliper diameter of a polyhedron as follows:
Let $P_1$ and $P_2$ be two planes both of which are parallel to the x axis such that the perpendicular distance between $P_1$ and $P_2$ is the ...
12
votes
3answers
554 views
General principles which lead to good questions in many concrete situations [closed]
I believe that in various fields of mathematics there are general principles which might lead to good questions and good results in many concrete situations. I would like to have a list of such ...
3
votes
1answer
98 views
Lattice projections
I imagine the following result is folklore
Theorem. Those $k$-dimensional subspaces $\zeta \subset \mathbb{R}^n$ $(1 \leq k \leq n-1)$ for which the orthogonal projection of the lattice $\mathbb{Z}^n$...
1
vote
3answers
193 views
Repeatedly halve and twist a planar shape: Limiting shape?
Consider the following iterative process.
Start with a planar region $R=R_0$ of $\mathbb{R}^2$.
I am thinking of $R$ as connected,
but it may become disconnected.
In the example below, $R$ starts as ...
5
votes
0answers
68 views
What quantum groups admit quantum topography space structure?
Quantum topography space is a pair $(A,M)$ consisting of a $C^*$-algebra $A$ and an abelian sub algebra $M\subset A$ with approximate identity. The intuition is to take $M$ be the smallest abelian ...
2
votes
1answer
150 views
About Vitali covering theorem
In the paper by Nazarov, Treil and Volberg: Weak type estimates and Cotlar inequalities for Calderon-Zygmund ... /1998, Int. Math. Res. Not.,
www.math.brown.edu/~treil/papers/l1/l1-5.pdf
on the ...
0
votes
1answer
156 views
Heat kernel and convergence
Let $(M_i,g_i,x_i)$ be a sequence of stochastically complete pointed Riemannanian manifolds ($x_i$ being the marked point on $M_i$) of injectivity radius uniformly bounded from below, Gromov-Haussdorf ...
3
votes
0answers
24 views
Tangencies of Villarceau circles in a 3D Steiner chain
Consider a Steiner chain made of an arbitrary number $n$ ($\geq 3$) of spheres (not circles, spheres), as in the picture below with $n=6$ (so it is a so-called Soddy hexlet). I've found this picture ...
6
votes
0answers
131 views
Approximating a ray with an integer lattice point
Take $X$ uniform on the unit sphere in $\mathbb{R}^n.$ For $r>0$, take $S_r=\{x\in \mathbb{Z}^n: \sum_i x_i^2 \leq r^2\}.$
With $\|\cdot \|$ the 2-norm, what is the distribution (or at least the ...
3
votes
1answer
76 views
Extremum problem on regular simplex
I'm looking for a proof for the extremum problem on regular simplex.
Question. Let $\mathcal{A}=A_0A_1...A_n$ be a regular simplex in $\Bbb E^n$. $P$ is a point inside and on boundary of $\mathcal{...
0
votes
0answers
41 views
Inequality on simplex with circumscribed sphere
I'm looking for a proof for this problem on simplex which I think it is true
Question. Let $\mathcal{A}=A_0A_1...A_n$ be a simplex in $\Bbb E^n$. $(S)$ is circumscribed sphere of $\mathcal{A}$ with ...
2
votes
2answers
106 views
Centroid and center circumscribed spheres in simplex
I'm looking for a proof for this problem on simplex which I think it is true
Question. $A_0A_1...A_n$ is a simplex in the Euclidean space $\Bbb E^n$. $G$ is its centroid and its center ...
2
votes
0answers
72 views
Euler line in metric space [closed]
If the answer about my question here is "yes" Coplanar set in metric space.
We shall have the concept of four coplanar points in metric space $(\Bbb M,d).$ I propose an idea about Euler line in ...
1
vote
0answers
70 views
Coplanar set in metric space
Let $(\Bbb M,d)$ be a metric space.
Give three points $X,$ $Y,$ $Z$ in $\Bbb M$ such that they satify one of the following conditions
$i)\ d(X,Y)+d(Y,Z)=d(X,Z),$
$ii)\ d(Y,Z)+d(Z,X)=d(Y,X),$
$iii)\...
7
votes
0answers
165 views
A conjecture on simplex
Let $A_0A_1...A_n$ be a simplex in $\Bbb E^n.$ Let $B_{ij}$ be a point on the edge $A_iA_j,\ 0\le i\not=j\le n.$
Denote by $\beta_i$ the hyperplane passing through the points $B_{i0},$ $B_{i1},$ $B_{...
5
votes
1answer
159 views
Cyclic quadrilateral in metric space
Consider a metric space $(\Bbb M,d).$
If $X,Y,Z\in \Bbb M.$ We define cosin of angle by
$$\cos(\angle YXZ)=\frac{d(X,Y)^2+d(X,Z)^2-d(Y,Z)^2}{2d(X,Y)\cdot d(X,Z)}.$$
If we have four points $A,$ $B,...
4
votes
1answer
104 views
Pascal's theorem for spherical hexagon
I draw a cyclic spherical hexagon and I check by geogebra that Pascal's theorem is true in this case.
My question 1. Is there simple proof for this?
My question 2. Can we change the circle on ...
4
votes
1answer
120 views
Coloring circles in plane
We assume that all the circles in the plane are each colored with one of two colors: red or blue.
My question 1. Does there always exist an equilateral triangle such that its circumcircle and its ...
3
votes
0answers
64 views
Miquel circles on sphere
Consider a sphere $\Bbb{S}$ on Euclide 3D space.
We well-known that a "line" connecting two points $X$ and $Y$ on $\Bbb{S}$ is the great circle of $\Bbb{S}$ which passes through points $X$ and $Y.$
...
1
vote
0answers
61 views
A generalization of the Sawayama-Thebault theorem?
Sawayama-Thebault theorem is one of the best nice theorem in Euclidean Geometry. The theorem is also has a long history. I had done above four years to find a generalization of this theorem. With many ...
4
votes
1answer
296 views
Converse of Pythagorean theorem in n-dimensional Euclidean space [closed]
We consider a simplex $A_0A_1...A_n$ in $n$-dimensional Euclidean $\Bbb E^n.$
Denote by
$Vol(A_0A_2A_3...A_n)=V_1,$
$Vol(A_0A_1A_3...A_n)=V_2,$
$...$
$Vol(A_0A_2A_3...A_{n-1})=V_n$
and
$Vol(...
15
votes
2answers
392 views
A cube is placed inside another cube
I known following problem with two square
$$Area(1)+Area(3)=Area(2)+Area(4).$$
My question. Is this problem true for two cubes?
We place a cube $XYZT.X'Y'Z'T$ into another cube $ABCD.A'B'C'D',$ ...
1
vote
0answers
52 views
Distance between quadric surface and point or Intersection of sphere and quadric surface
I asked a similar question on math.stackexchange, but the answer wasn't quite ideal for my application. Apparently analytic solutions are surprisingly rare for general quadric distances.
Given a ...
1
vote
0answers
114 views
$N$-$th$ chain of six circles
Since 2013, I found a very nice configuration: $N$-th chain of six circle. This is a generalization of theorem 1 in here and theorem 2 in here (and is also generalization of Pascal theorem even with $...
8
votes
1answer
184 views
Bi-Lipschitz extension
Given a bi-Lipschitz homeomorphism
$\Phi:\mathbb{B}^n(0,1)\to\mathbb{R}^n$, (that is a bi-Lipschitz map onto the image), can one find a bi-Lipschitz homeomorphism $\Psi:\mathbb{R}^n\to\mathbb{R}^n$ ...
2
votes
0answers
93 views
Pascal theorem for three dimensions
A year ago I found the Pascal theorem for three dimentions as follows:
Let $(C_1)$, $(C_2)$ be two conics on the same Ellipsoid, (or Hyperboloid, or Paraboloid). Let $A_1$, $A_2$, $A_3$, $A_4$, $A_5$,...
