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Euclidean, hyperbolic, discrete, convex, coarse geometry, comparisons in Riemannian geometry, symmetric spaces.

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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 ...
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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
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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 $\varepsilon \in (0,\varepsilon_0)$...
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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 ...
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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
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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
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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 $...
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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? ...
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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 ...
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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
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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 $\...
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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 (...
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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
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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? ...
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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
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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
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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 $B\subset G$ ...
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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 $\{X_i^n\}_{i\in \mathbb{N}}$ (...
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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, ...
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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 $\theta_1\sim \operatorname{Uniform}(t_1,t_2)$, $\theta_2\sim \operatorname{Uniform}(t_3,t_4)$,...
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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 ...
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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-...
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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 ...
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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
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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
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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
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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 $\mathbb{Z}^n$...
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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
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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
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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 ...
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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
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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 ...
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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
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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{...
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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 ...
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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
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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 ...
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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)\...
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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
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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
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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
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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 ...
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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 $Y.$ ...
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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
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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(...
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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 $ABCD.A'B'C'D',$ ...
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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 ...
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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
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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 $\Psi:\mathbb{R}^n\to\mathbb{R}^n$ ...
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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 $A_1$, $A_2$, $A_3$, $A_4$, $A_5$,...