Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,404 questions
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Two questions about convex subsets of Hilbert Space
Let H be an infinite dimensional and separable Hilbert Space. Do there exist disjoint, closed and bounded subsets A,B of H which satisfy the following conditions? (1) Each of A,B is convex and has a ...
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107
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Constructing measures with support in a given set
I've recently come across the Frostmann Lemma (http://en.wikipedia.org/wiki/Frostman_lemma). Its proof involves constructing a measure with certain properties on a given subset of $\mathbb{R}^n$ (I'm ...
- 4,727
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201
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harmonic forms with respect to different metrics
Given a smooth manifold M with two different metric, we can consider U and V, the spaces of harmonic k-forms on M with respect to the first metric and second metric respectively. The question is ...
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Proper Group action on a metric space
Let $(X,d)$ be a metric space and $C\subset X$ be a compact subset. Let furthermore $G$ be a group that acts on $X$ proper and by isometries. Does there exist an $\epsilon >0 $ such that: Let $U=$ {...
- 1
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604
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Finding lattice with short basis-vectors containing given lattice
While working on understanding the space spanned by certain integer relations of real numbers I have come across the following problem. Given $v_1,\dots, v_n \in \mathbb{Z}^m$, I am would like to find ...
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526
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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
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28
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Can the Pythogrean theorem be proved using imaginary numbers?
Can the Pythogrean theorem be proved using imaginary numbers? The proof must avoid circular reasoning, of course.
I asked essentially the same question at MSE, but did not receive a definitive answer, ...
- 3,527
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90
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How to calculate the maximum dimensions of a rectangle inside two concentric circles? [closed]
If I have a rectangle ABCD such that A and B touch two points of the outer circle and CD's touches one point of the inner circle, how could the maximum dimensions of the rectangle be calculated?
...
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114
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Geometric interpretation of a Grammian-like function
Let $\mathbf{v}, \mathbf{w} \in \mathbb{R}^n$ and consider the following function $f : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$:
$$
f(\mathbf{v},\mathbf{w}) = \|\mathbf{v}\|\|\mathbf{w}...
- 6,351
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37
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L2 distance computation with given distance to triangle nodes [closed]
In a triangle with three points A, B, and C. The L2 distance between each pair of points |AB|, |AC|, |BC| is given. For the other two points O and P, the distance to the three points is given, i.e. |...
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Name this geometric point?
Is there a formal name for the point which is the reflection of the incenter about the circumcenter of a triangle?
- 1,109
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199
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Leech lattice shortest vector vs other 23 cases and E8 cases
In this paper by Viazovska, she said that:
"The E8-lattice sphere packing 𝒫E8 is the union of open Euclidean balls with centers at
the lattice points and radius $1/\sqrt{2}$." So I think ...
- 447
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247
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Projecting a given point onto a random $2$-dimensional plane in more than $3$ dimensions
We are given $\mathbf{p}\in\mathbb{R}^d$, where $d\gg 1$. Let $\mathbf{v}$ be a point selected uniformly at random from the unit $(d-1)$-sphere $\mathcal{S}^{d-1}$ centered at the origin $\mathbf{0}\...
- 1,677
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Geometry problem about finite set in unit ball
I think covering numbers (of the unit ball) is the right way but I think about this question for a few days now and a hint would by nice….
Let $B_{1}$ be the unit ball in $\mathbb R^d$ and let $A \...
user475607
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445
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Standard Gram matrices for lattices
I would like to define standard Gram matrices, and use them to help me understand the symmetries of lattices.
I define "standard Gram matrix" as the Gram matrix g that minimizes the ...
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209
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Question about Clifford multiplication
Let $X$ be a smooth vector field on the even dimensional sphere $S^n$. Let $S(TS^n)=S^+(TS^n)\oplus S^-(TS^n)$ be the spinor bundle over $S^n$ equipped with a bundle metric that is compatible with the ...
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Cauchy's rigidity theorem
Newbie here, I'm studying the proof of Cauchy's rigidity theorem, but couldn't find any good resources. I read the chapter about it Proofs from THE BOOK, but it's really brief and I was not able to ...
- 31
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99
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What is sequential boundary of a $\delta$-hyperbolic space and how is the Gromov product extended to the boundary?
I have been reading up on $\delta$-hyperbolic spaces. But I am not getting a clear idea of sequential boundary of $\delta$-hyperbolic spaces and how the Gromov product is extended to it. Could ...
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Maximal families of equal length intervals consist of equilateral triangles
My question is a follow up to How to find n points on a plane so that as many pair of points as possible have the same distance? -- see the conjecture at the bottom of this post.
Let $\ n\ $ be a ...
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Lower Estimate of A Lipschitz Map
Suppose that $(X,d_X)$ and $(Y,d_Y)$ are complete doubling metric spaces and let $f:X\rightarrow Y$ be a non-constant Lipschitz map. Then can does there exist a lsc function
$\rho:(0,\infty)\...
- 5,405
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79
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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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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 ...
user50806
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524
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Compose/decompose rotation matrix from/to plane of rotation and angle
I would like to compose/decompose an $n$-dimensional orthogonal rotation matrix (restricting to simple planar rotations, which rotates in the specified plane of rotation, and fixes in the plane ...
- 111
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71
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A bound on the Haussdorff distance
Let $X, Y \subset \mathbb{Z}^2$ be two discrete and bounded sets. Let $f_X$ be the Euclidean signed distance function of $X$ (similarly for $Y$) and $d_H(X,Y)$ the Euclidean Haussdorff distance ...
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harmonic maps from cone to $S^2$ locally lipschitz?
Are the harmonic maps from a 2-dimensional cone to $S^2$ locally lipschitz or Holder continuous?
- 689
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376
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Probability of disc-disc overlap for discs placed with uniform probability on a surface until a density $\rho$ is achieved
Imagine I place discs of radius $r$ on a two-dimensional plane, selecting their positions with uniform probability across the surface of the plane, and stop when I reach a disc density $\rho$. As a ...
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208
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The pth power of a distance function is twice continuously differentiable, for $p>2$?
Suppose $\mathcal{O}$ is an open convex connected strict subset in $\mathbb{R}^n$ and define $\beta(x)=dist(x, \mathcal{O})$, for each $x\in\mathbb{R}^n$.
Is $\beta^p$, $p>2$ a twice continuously ...
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393
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Why is the physical space equivalent to $\mathbb{R}^3$ [closed]
I am trying to understand what would be the logical reasons behind our assumption that our physical space is equivalent to $\mathbb{R}^3$ or 'physical straight line' is equivalent to $\mathbb{R}$.
$\...
- 1,305
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Shortest distance along the surface of the hyperboloid [closed]
Given two points A and B on the surface of the hyperboloid x^2+y^2-z^2=1. How to find the shortest distance between them along the surface?
- 109
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Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry? [closed]
In an Euclidean plane, we know that the area of a triangle is determined by the length of base and the height, i.e.,
$$
S_{\Delta}=\frac{1}{2}a.h,
$$
where $a$ is the length of base and the $h$ is ...
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376
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Mean Three Dimensional Shape of Surfaces
If I have $n, 1 < i < n, $ surfaces composed of $f_i$ faces and $v_i$ vertices, how would I go about finding the average surface?
(I'm unsure what I mean by average - intuitively it's obvious, ...
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333
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Explicit example of a smooth - but not analytic- closed curve without self-intersections
There exist smooth - but not analytic - closed curves without self-intersections. I just would like to see a simple example of such a curve.
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Polygon Chain - Conversion to non-crossing while preserving shape?
I have polygon chains similar to the following...
http://upload.wikimedia.org/wikipedia/commons/thumb/6/62/Self_crossed_polygonal_chain.svg/220px-Self_crossed_polygonal_chain.svg.png
...given the ...
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Metric for measuring linearity of finite set of points in $R^2$
Suppose one has $n > 2$ points in $R^2$, and one wants to measure "how linear" they are.
I want a metric such that (a) if all the points are in fact on the same line, the metric gives 1, (...
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131
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Is this a smooth approximation to the $\ell$-infinity distance actually a quasi-metric?
The $\|\cdot\|_{\infty}$-norm on $\mathbb{R}^n$ for $n\in \mathbb{Z}^+$ is not a smooth function. However, I came across this post which essentially says that a pointwise approximation to the maximum ...
- 169
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$\omega$-homogenous space which is not $\omega_1$-homogenous
Consider a metric space $(X,d)$ and let $\kappa$ be a cardinal. We say that $(X,d)$ is $\kappa$-homogenous, if every (surjective) isometry $h:X_1 \to X_2$ between subspaces of $(X,d)$ of size $< \...
- 135
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Curvature of space of paths into a NPC space
Let $(X,d)$ be a complete separable geodesic (thus connected and path connected) metric space of non-positive curvature (in the sense of Ballmann) and fix some $x_0\in X$. Let $C_0(I,X)$ denote the ...
- 364
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Invariance of Minkowski sum of sets
Given an euclidean space $E$, two sets $A,B\subset E$ and the action on $E$ of two groups $G_A,G_B$ such that $G_A A=A$ and $G_B B=B$, it is possible to generate a group that leaves invariant $A\oplus ...
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Estimation via projecting onto a convex body
Assume that $\theta$ is in a convex body $K \in \mathbb{R}^n$ and we observe $y = \theta + z$, where $z$ is a noise term (following, say, the normal distribution). Consider an estimator of $\theta$ by ...
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Distances from a parallelogram to a plane
Consider a parallelogram $ABCD$ and a plane, $P$. Let $A'B'C'D'$ be the orthogonal projective image of $ABCD$ onto $P$. If $P$ cuts the segments $AC$ and $CD$, then $BB'=AA'+CC'+DD'$. If $P$ cuts the ...
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146
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Upper bound of Wasserstein distance given by subvariables of codim 1
recently I am considering the upper-bound of Wasserstain distance. Say we have random vectors $X,Y$ of dimension $n$, and let $\tilde{X}_i (\tilde{Y}_i,$ resp.) be the $(n-1)$-dim random vector of $X (...
- 363
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intersection and self intersection on non compact complex surfaces
I'd like to have a definition of intersection on non compact complex surfaces because all i have found so far is only about projective surfaces. for example how can i define the self intersection of ...
- 25
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Fast way to generate random points in 2D according to a density function
I'm looking for a fast way to generate random points in 2D according to a given 2D density function.
For instance something like this:
Right now I'm using a modified version of "Poisson disc&...
- 121
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223
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Dense $G_{\delta}$ set with $\sigma$-porous complement is cofinite?
Let $X$ be a separable Banach space and $D\subseteq X$ be a
proper, connected, and dense $G_{\delta}$ subset of $X$,
$X-D$ is $\sigma$-porous.
Then is $X-D$ contained in a finite-dimensional ...
- 5,405
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Does every compact doubling metric space have a canonical measure?
My question is this one, with the additional condition that the metric space be doubling. In the aforementioned question, the limiting measure depends on the sequence $\epsilon_n$ and hence is not ...
- 6,541
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What’s the form of Gram matrix for right-angled hexagon
Informally, right-angled hyperbolic hexagon is a hyperbolic triangle with vertices outside infinity. I think there should be a Gram matrix for it, and what does it looks like?
(The Gram matrix here ...
- 21
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243
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Covering numbers of uniformly bounded subsets of Gromov-Hausdorff space
For any metric space $X$ and $\varepsilon>0$, let $$\text{cov}(X,\varepsilon)=\min\{n\,|\,X\text{ has a cover by }n\text{ many closed }\varepsilon\text{-balls}\},$$
be the ordinary covering ...
- 12.5k
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395
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Showing convexity of a function in the unit ball
We have the unit sphere $S^2$ in $\mathbb{R}^3$ and two points, $X$ and $Y$ on the surface of the sphere. Then, a function is defined for any point $P$ inside of the unit ball as:
$$f(P) = R\,d(P, XY)...
- 3
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96
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Optimal covering with finite subcollection of open sets
This is mainly a reference request. Consider a finite collection of (let's say, for simplicity) of open balls $B_i, i = 1, 2, ..., m$ in (again, for simplicity) $\mathbb{R}^n$. I am looking for ...
