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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Metric / strong slope restriction of function on unit ball in $\mathbb R^m$
Diclaimer. I'm not sure this is the right venue for this question, but I'll give it a try
Definition [Strong / metric slope]. Given a complete metric space $(M,d)$ and a function $f:M \to (-\infty,+\...
- 6,853
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113
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Folding a non-rectangular shape into a rectangle of uniform thickness
I think the following might be an interesting subproblem of this question:
Question: For an odd number $n\ge 3$, is there a non-rectangular but still convex shape of area $A=1$, that can be folded (...
- 13.6k
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How to find the coordinates of these points?
Does anyone know any way or any algorithm that can exactly and/or numerically find the coordinates
of $n_{k}+2$ equally spaced points on the $(n_{k}-1)$-dimensional
unit sphere $S^{n_{k}-1}$ for some ...
- 19
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290
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When is the internal covering number of a metric space monotonic?
Given a radius $r > 0$, the internal covering number of a subset $T$ of a metric space $(X, d)$ is denoted $N_r(T)$ and is defined to be the smallest number of balls of radius $r$ (under $d$) with ...
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237
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A basic question about compact $C^1$ surfaces with boundary
Let $S \subset \mathbb{R}^3$ be a compact and locally $C^1$ simply-connected surface with a $C^1$ boundary with no self intersection. Is there a $C^1$ bijection $F: \overline{B(0,1)} \rightarrow \...
- 434
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150
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A Uniform Metric Selection Theorem
Let $X$ and $Y$ be bounded complete separable metric spaces. Let $C = 2^\omega$ be Cantor space with its standard metric. All product spaces are taken to have the max metric.
Let $F, G \subseteq X\...
- 12.5k
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96
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Parallel hyperplane to a hyperbolic isometry of a CAT(0)-cube complex
Consider a finite-dimensional geodesically complete CAT(0) cube complex $X$.
A hyperplane of $X$ is a convex subspace of $X$ that intersects the mid-point of edges of $X$ such that $X - H$ has two ...
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242
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Barycenter Map on Wasserstein Space
Let $(X,d)$ be a complete separable metric space, $P_1(X,d)$ be the set of Radon probability measures on $X$ satisfying
$$
P_1(X,d)\triangleq \left\{
\nu:\,(\exists x_0\in X)\, \int_{x\in X} d(x,x_0)d\...
- 5,405
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Reference request: metric spaces with curvature bounded from below (CBB) spaces
What is the/a main reference book for spaces with curvature bounded from below (CBB spaces/spaces with curvature $\geq \kappa$ in the sense of Alexandrov)? Looking for an up to date reference.
- 447
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452
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Topology of length spaces
How wild can the topology of a length space be? That is,
Questions:
Let $X$ be a metric space where the distance between two points $x,y \in X$ is the infinum of lengths of rectifiable paths from $...
- 63.9k
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Formally proving that a metric is not induced by any norm in $\mathbb{R}^n$ [closed]
What is the procedure to formally prove that no norm exists in $\mathbb{R}^n$, that induces a metric $d$?
My first instinctive idea would be to show that $d$ is a metric in $\mathbb{R}^n$, but after ...
user138015
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Metric 1-current decomposition
I've been reading Paolini-Stepanov arcticle and in section 4, at page 6, they define a metric current from a transport:
$$T_{\eta}(\omega)=\int_{\Theta}[[\theta]](\omega)d\eta(\theta),$$
which ...
- 391
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Is a polytope with vertices on a sphere and all edges of same length already rigid?
Let's say $P\subset\Bbb R^d$ is some convex polytope with the following two properties:
all vertices are on a common sphere.
all edges are of the same length.
I suspect that such a polytope is ...
- 13.6k
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589
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Valid metric on a hyperbolic space
Note: originally posted on math.SE.
I'm looking at the distance that's defined in this paper on Poincaré Embeddings:
$d(\mathbf{u}, \mathbf{v}) = \operatorname{arccosh} \left(1 + 2\frac{\left\| \...
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Growth rate of bounded Lipschitz functions on compact finite-dimensional space
Let $\mathcal X$ be a metric space of diameter $D$ and "dimension" (e.g doubling dimension) $d$. Let $L \in [0, \infty]$ and $M \in [0, \infty)$ and consider the class $\mathcal H_{M,L}$ of $L$-...
- 6,853
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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, \...
- 1,776
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For what points in $\mathbb R^2$ does the triangular condition fail for a 1/2-metric?
It is known that for $p<1$, $d_p(x,y)=\big(\sum_{i=1}^n |y_i-x_i|^p\big)^{1/p}$ is not a metric. In the case of 2 dimensions and $p=1/2$ it seems rather hard to find a counterexample where the ...
- 1,465
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How does Siegel's Hilbert-Blumenthal fundamental domain differ from Götsky's?
This question has been changed to something related but different from the original question. Thanks to @paulgarrett for chatting with me and helping me hone in on a more interesting part.
The first ...
- 2,436
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218
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From measurable to quantitative estimates of a map in the coarea formula
Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m $ be Lipschitz and $n \geq m$. A version of the coarea formula says:
$$ \int_A g(x) J_m f(x) d \mathcal{L}^n (x) = \int_{\mathbb{R}^m } \int_{ A \cap f^{-...
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About pairwise distances of some points in a Riemannian manifold $M$ of ${\rm sec}\ M\geq 1$
This question is cross-posted in MO and MSE https://math.stackexchange.com/questions/2276064/about-pairwise-distances-of-some-points-in-a-riemannian-manifold-m-of-rm-se
Assume that there are points ...
- 1,100
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148
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Dimension of quotient of compact totally disconnected group action
Assume that $X$ is a compact metric space and $G$ is compact
totally disconnected group. And $X$ has isometric free $G$-action
i.e. $gx=x\Rightarrow g=e$.
Then the following holds $${\rm dim}\ ...
- 1,100
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255
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Determining the rate of spread of geodesics when the sectional curvature is zero
I have posted this question in mathSE a few weeks ago (and proposed a bounty) but so far got no response.
In the book Riemanian geometry (by do-Carmo), the following result is proved (Corollary 2.9 ...
- 6,741
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273
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Regularity of a generalized polar coordinate metric with two angles
Flat space in polar coordinates takes the form
$$ds^2=dr^2+r^2d\phi^2$$
To avoid a conical singularity at the origin, we must impose that $\phi$ is periodic with period $2\pi$.
Now consider the ...
- 293
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Enclosing a convex plane domain in a disc
The following statement seems obvious to me:
Let $\gamma:S^1\to\mathbb R^2$ be a smooth injection such that $\dot\gamma$ and $\ddot\gamma$ never vanish.
Then $\gamma$ encloses a strictly convex ...
- 8,086
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203
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Descartes' theorem and Circle Packing [closed]
There's something I am missing comparing Descartes' theorem for three isometric circles here and this wiki post on circle packing of 3 circles here.
From my calculation:
$$
r_{ext} = \frac{r_{int}}{{...
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284
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The pointwise Lipschitz-ness of a function on a dense set, implies its pointwise Lipschitz-ness everywhere?
Some definitions: Let $(M,d)$, $(M',d')$ be metric spaces. For $f:M\to M'$, $x\in M$ and $r>0$, define $$D_r(f)(x):= \sup\{r^{-1}d'(f(x),f(y)): y\in M,\,d(x,y)\leq r\}.$$ Define the pointwise ...
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145
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Continuity of Busemann-Hausdorff area density
I am trying to find out why the Busemann-Hausdorff area density as defined by Burago and Ivanov is continuous. Here, $GC_m(V)\subset \Lambda^m(V)$ denotes the simple $m$-vectors in an $n$-dimensional ...
- 193
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Multiplicity of a subcovering in spaces of given Hausdorff dimension
Let $X$ be a locally compact metric space of integer Hausdorff dimension $n$. Let $K\subset X$ be a compact subset. Let $\{B_i\}_i$ be a finite family of balls covering $K$. One may assume that all ...
- 21.8k
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Pairs of rays in euclidean buildings
In section 4.1.3 of Kleiner and Leeb's paper on symmetric spaces and euclidean buildings, there's a result about pairs of rays from the same point initially spanning a flat triangle (or being ...
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Extreme points of convex hull of Minkowski sum [closed]
Let $\operatorname{conv}(a_1,\ldots,a_m)$ denote the convex hull of $\{a_1,\ldots,a_n\}$. Let $P = \operatorname{conv}(a_1,\ldots,a_p)$ and $Q = \operatorname{conv}(b_1,\ldots,b_q)$ be two convex sets ...
- 1,713
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Existence of shortest paths in complete Alexandrov spaces
Let $X$ be complete finite dimensional Alexandrov space with curvature bounded from below. Is it true that any two points can be connected by a shortest path? If this is not true in general, it it ...
- 21.8k
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2
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310
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Are there CAT(-1) spaces which are not trees whose Gromov boundary is disconnected?
Are there some examples of CAT(-1) spaces which are not trees which have disconnected Gromov boundary?
- 2,964
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Smooth unit vector field on a tetrahedron to interpolate vertex constraints
For a tetrahedron $T\subset \mathbb{R}^3$ with vertices $r_i\in \mathbb{R}^3$ , $i=1,\ldots,4$, and unit vectors $u_i\in \mathbb{S}^2$ at each vertex $i=1,\ldots,4$
consider the (energy) functional
$$...
- 1,676
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475
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Sampling uniformly from all possible line segments of a given length that fit inside a container
Consider that task of randomly placing a line segment of some length $L$ near a plane s.t. a point $p$ at the center of the line segment is at most a distance $H$ from the plane and intersections ...
- 11
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1
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925
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Normal tubular neighborhood theorem for semi(or pseudo)-riemannian manifolds
Suppose you have a manifold $M$ and a closed sub-manifold $A$, and let $g$ be a semi-riemannian metric,ie, $g_x$ defines a quadratic form on $T_xM$ such that $g_x(v,v)\ge0$, but $g_x(v,v)=0$ not ...
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1
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680
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Harmonic function defined on a cone
It's well known that: Given a continuous function defined on the boundary of the disk, then there exists a unique harmonic function in the interior of the disk. What if we replace the disk by a cone?
...
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heat kernel $p_t(x_0,y) \in D(\Delta) \cap L^\infty$ for a manifold with Ricci curvature bounded below?
X is an n-dim Riemannian manifold with the Dirichlet form
$$
\varepsilon (u,v) =-\int_X \langle \nabla u,\nabla v \rangle
$$
for $u,v \in W^{1,2}(X)$.
Let $P_t$ and $p_t(x,y)$ be the associate ...
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587
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examples where some point of the Gromov-Hausdorff limit space has non-unique tangent cones
Suppose $\left( {{M_i},{q_i}} \right)\mathop \to \limits^{G - H} \left( {X,{q_\infty }} \right)$, ${\rm Ric}_{M_i} \ge - \left( {n - 1} \right)$, and ${q_\infty }$ is regular, i.e. all the tangent ...
- 689
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2
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Smoothness and curvature of geodesics in a length space
Let $X$ be a nice compact subset of $R^d$. Given a function $p: X \to \mathbb{R}^+$, define the length of a path $\gamma \subset X$ as $\ell(\gamma) = \int_\gamma p \, ds$, and the distance between ...
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Fractal dimension of 1D set, what if logN vs log(e) is a polygonal chain?
I have a finite set of points, and plot the graph log(N) vs. log(e). I see a polygonal chain (the final slope, starting at some size of e, is zero, of course). If the set represents some physical ...
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Is this min not less than a min
Let $\mathbf{D}$ be the unit disk, is
$$\inf_{\begin{array}{c}
v_{1},v_{2},v_{3},v_{4}\in\mathbf{D},\\
v_{0}\in\mbox{convexhull}\left(v_{1},v_{2},v_{3},v_{4}\right)
\end{array}}\max_{0\le i,j,k\le4}\...
- 27
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3
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378
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Surface integral approximation
Let $D$ be a bounded Lipschitz domain and $f$ is continuous up to $\partial D$. Is it true that $$\int_{\partial D}f(x)d\sigma(x) = \lim_{\epsilon\to 0}\frac{1}{\epsilon}\int_{D^{\epsilon}}f(x)dx$$
...
- 113
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763
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Existence of cut-off functions in metric spaces
Let $X_1,\ldots,X_m$ be Lipchitz continuous vector fields in an open set $\Omega \subset \Re^n$, Let $d(\cdot, \cdot)$ denote the control distance associated to $X$. With respect to this control ...
- 116
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2
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339
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Smooth a matrix
I have a matrix in which each element contains the coordinates of a 3D surface. Sometimes, some points will be "out of line" meaning that they will not conform to the general shape. For example you ...
- 87
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1
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233
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s-densities and conformal measures
I recently learnt about s-densities:
http://en.wikipedia.org/wiki/Density_on_a_manifold#s-densities_on_a_vector_space
For simplicity suppose that the vector space in this definition is \R. The prime ...
- 11
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1
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698
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Number of connected components of complement to a reducible real algebraic hypersurface.[EDITED]
Let $X_1,\ldots X_k$ be irreducible(may be singular) affine real algebraic hypersurfaces in $R^n$ with $x_1,\ldots, x_k$ connected components, respectively.
Let $G_1,\ldots, G_l$ be their ...
- 413
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1
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227
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Constant hole density on the area of a circle
I need to create about 100 (small) holes in a distributor plate (hole diam = 0.5 mm; plate diameter = 100 mm). The sm. holes should be distributed in such a way that the density (hole/area) is nearly ...
- 13
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1
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266
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Warped product neighborhoods of geodesic hypersurfaces
Is it true that any totally geodesic hypersurface in a nonpositively curved manifold has a tubular neighborhood such that the metric on the neighborhood is a warped product?
At least, if the manifold ...
- 73
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1
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283
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Constructing a graph that approximates a sphere using rotationally symmetric building blocks with equal numbers of edges
I'd like to construct a graph that approximates a sphere in 3-space, but I'm placed under the following constraints:
(1) - I am only allowed to use a construction block, $v_i$, consisting of a single ...
