Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,403 questions
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Multi-layered wrapping of polyhedra
This post continues from How big a box can you wrap with a given polygon? and Convex polyhedra that can be folded from convex polygons. One can also mention 'k-fold coverings of the plane' as examined ...
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Essential singular set of an Alexandrov space
Let $X$ be a locally compact Alexandrov space with curvature bounded below. Suppose $C$ is a closed subset that consists of the essential singular points, where a point $p$ is called an essential ...
- 891
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Density of $C^k$-functions with Lipschitz partial derivatives
Let $N$ and $M$ be complete Riemannian manifolds, of respective dimension $n$ and $m$ with $n,m\geq 1$. Let $C^{k,1}_b(N,M)$ be set of all bounded continuous functions $f:N\rightarrow M$ for which ...
- 5,405
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Further queries on inside-out polygonal dissections
The following is based on Inside-out polygonal dissections
Definition: We say that a polygon P has an inside-out dissection into another polygon P' if P′ is congruent to P, and the perimeter of P ...
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In-plane right Cauchy Green tensor for a 2D manifold
Consider a surface in a 3D Cartesian frame undergoing a deformation - I'm looking for a derivation of the right Cauchy Green tensor (the metric of the deformation) in the tangent plane of the surface.
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What do you call this notion of similarity between subsets of a metric space?
Given a metric space $X$, we say a continuous isometry is a continuous map $F:X\times[0,1]\to X$ such that defining $f_t(x) = F(x,t)$ for $t \in [0,1]$, we have that $f_0$ is the identity map and $f_t$...
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A generic question on circles associated with a triangle
This question is inspired by two posed by P.Terzić (both given elegant synthetic proofs by F. Petrov). The starting point is a triangle $ABC$ and a triangle centre $G_1$. There are two classical ...
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Does there exist an isometry between a regular polygon and a circle?
In order to define the question in a meaningful fashion, I am referring to a smooth manifold $\mathcal{M}$ within an $\epsilon$-neighborhood of a regular polygon $\mathcal{P}$ satisfying $$\max\{\|x-p\...
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Deployment and dispersion in triangular regions
Definitions (from C. Stanley Ogilvy's 'Tomorrow's Math'):
Deployment: To place a specified number $n$ of points (stations) in a region such that the maximum distance of any point in the region from ...
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Computing/estimating geodesics in practice
Let's say I have a Riemannian manifold with known metric $g$. I want to compute the geodesics of this manifold, specifically with respect to the Levi-Civita connection.
In practice, (i.e. with a ...
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Actions of finite groups on compact symmetric spaces
I am interested in series of finite subgroups of the classical compact simple Lie groups which have big orbits on compact symmetric spaces and where the double coset space has some nice explicit ...
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Classification of pseudoregular polyhedra
In contrast to a regular polyhedron, which has one orbit of flags, I’ve been studying what I call pseudoregular polyhedra, which have two orbits of flags interchanged by conjugation (explained here). ...
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Reference request: a class of matrices leading to interesting metric geometry
For $0 \le A \in GL(n,\mathbb{R})$, let $Aw = \Delta(A)$, where $\Delta$ denotes the map taking a matrix to a vector of its diagonal entries and/or forming a diagonal matrix from a vector, according ...
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Does the Hausdorff dimension characterise CAT(0) spaces having some bilipschitz balls?
It is well-known that the Hausdorff dimension is invariant under bi-Lipschitz mappings. I would be interested in a specific converse of this invariance. Let $X$ and $Y$ be two CAT(0) spaces having the ...
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Number of lattice points in a structural symmetric convex body
Let $f$ is a convex symmetric function on the interval $[-a,a]$, i.e., $f(-x)=f(x)$ for $\forall \, x\in [-a,a]$. Then we consider a $n$-dimensional convex body in Euclidean space
\begin{equation}
\...
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Metric transforms that preserve $\ell^1$ embeddability
Consider a function $f$ from reals to reals such that $f$, when applied to pairwise Manhattan distances between $n$ points, always results in a set of Manhattan distances.
Work by Schoenberg and ...
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Constructive way to optimally cover a compact subset of Euclidean space
Let, $(X,d)$ be a simply connected compact subset of $\mathbb{R}^d$ with non-empty interiorn, let $d$ denote the Euclidean metric, and let $\varepsilon>0$. Is there a way to iteratively select ...
- 5,405
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Bound for the cardinality of maximal $r$-separable subsets contained in a ball of radius $R$ in $\mathbb R^d$
Let $B$ be a closed ball in $\mathbb R^d$ of radius $R$ and let $N=N_R(r)$ denote the maximal cardinality of the $r$-separated sets (meaning any two points in this set have distance at least $r$) that ...
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Is the mean center of the vertices of a convex polygon always inside the polygon? [closed]
As simple as that. I'm doing an R program where I need to order clockwise a bunch of points that describe a regular polygon and to do that I figured I could find a point inside, change to polar ...
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Recursive expression of Lebesgue measure for balls with removed intersection
This is not the most theoretically challenging question; rather it is more of a reference request for a simple formula (which must be known).
Let $\left\{B_{\epsilon_n}(x_n)\right\}_{n=0}^N$ be a set ...
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Lax CD(K, $\infty)$ space in the sense of Sturm
In K.T. Sturm's "On the geometry of metric measure spaces. I", Definition 4.5, he introduces a "lax" version of the usual CD(K,$\infty$) lower bound. Namely, one allows for $\...
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Neighbor count in sphere packing in N dimensions
So I'm really interested in building a mathematical model for how powerful computer chips could be given extra spatial dimensions. Obviously this is a squishy problem, since "computer chips" ...
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If the volume-ratio of an inscribed convex set to the circumscribing convex set is rational, can anything of consequence be further deduced?
Say, one has two $n$-dimensional convex sets $A$ and $B$, with $B$ being inscribed in the strictly larger set $A$. ($A$ and $B$ have at least one boundary point in common. $B$ “fits snugly” in $A$ ...
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When is inverse geodesic distance positive definite (in a compact manifold)?
We work on a closed smooth Riemannian manifold $(M,g)$ and let $K:M\times M\to \mathbb R\cup\{+\infty\}$ be a kernel, which we assume to be integrable and lower semicontinuous. We say it is positive ...
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What is the maximum number of points on a plane such that some other point A is the nearest neighbor for all of them? [closed]
Point $A$ on the plane is given. How many other points can be placed on this plane under the condition that $A$ is the nearest neighbor for any of these points?
I can think of a regular pentagon with $...
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Minkowski (box-counting) dimension of generalized Cantor set
I'm trying to solve this problem.
For $0<\alpha, \beta<1,$ let $K_{\alpha, \beta}$ be the Cantor set obtained as an intersection of the following nested compact sets. $K_{\alpha, \beta}^{0}=[0,...
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Continuity of conformal grafting, wrt the (infinite type) surface
Say I have two closed hyperbolic surfaces $X,Y$ and a smooth, $(1+\epsilon)$-bilipschitz map $f : X \to Y$ for some small $\epsilon$. Pick a simple closed curve $c \subset X$, and let $X',Y'$ be the ...
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How can construct the equilateral $A''B''C''$ such that area of $A''B''C''$ is biggest
Let $ABC$ be arbitrary triangle in a plane. Let $A'B'C'$ and $A''B''C''$ be two equilateral triangles such that $A \in B'C'$, $B \in C'A'$, $C \in A'B'$ and $A \in B''C''$, $B \in C''A''$, $C \in A''B'...
- 1,776
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n-dimensional polyhedron with special properties
I'd like to know if there exists a convex face transitive n-dimensional polyhedron with all dihedral angles equal to $\frac{2\pi}{3}$.
For n = 2,3,4 an example can be a regular hexagon, a rhombic ...
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Gromov-Hausdorff limit of compact surfaces with same boundary of equal areas
Define $$ A = (0,0,0), \ B=\bigg(\frac{1}{n},0,0\bigg), \ C=
\bigg(\frac{1}{n},\frac{1}{n},0\bigg) ,\ D= \bigg(
0,\frac{1}{n},0\bigg),\ E=
\bigg(\frac{1}{2n},\frac{1}{2n}, \frac{a}{n}\bigg)
$$ for ...
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How dense can a transitive sets of points be?
How dense can a finite set of points on the $d$-dimensional unit sphere be if I require that the symmetry group of that arrangement is still transitive on the points?
As a measure for density I use ...
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Can an orderless set of inner product between N vectors determine unique structure of the vectors?
Suppose we have n vectors {a1,a2,a3,...,an} such that the sum of them is zero vector
a1+a2+a3+...+an=0
Now, we compute the inner product of each two vectors of them, i.e. we compute the Gramian matrix ...
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Euclidean embedding of the mesh
$M$ is a topological mesh, i.e. triple $M=(V,E,F)$, where $V$ is the vertex, $E$ is the edge and $F$ is the face, such that $M$ is homeomorphic to the sphere.
Suppose that we have a metric $l :E\...
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Bound between distance between Rotation Matrices
Let $\|\cdot\|_F$ denote the Fröbenius norm on the set of $d\times d$ matrices. By restriction this induces a metric on $SO(n)$.
Let's make an observation.
Since $X\in SO(n)$ is a rotation matrix ...
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Determining the behavior of a contraction mapping with undefined points
Label $X$ as the real interval $[0, a]$ where $a \in \mathbb{R}^+$, so that $\text{int}(X) = (0, a)$ labels the interior of $X$ and $\partial X$ labels the boundary of $X$. I have a function $f:\text{...
- 1,087
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Gromov-Hausdorff distance between graphs with edges as part of the space versus not part of the space
Let $G_1$ and $G_2$ be finite simple graphs viewed as metric spaces in the natural way where the edges are not part of the space. Let $G_1'$ and $G_2'$ be copies of $G_1$ and $G_2$ resp. but with the ...
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Associativity property of the gyrobarycenter
I'm using Ungar's terminology and notations. In the open unit ball of $\mathbb{R}^n$, let $GB(A_1, \ldots, A_N; m_1, \ldots, m_N)$ be the gyrobarycenter of the points $(A_1, \ldots, A_N)$ with ...
- 2,560
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Metrics on the space of distributions in terms of p.d.fs
If two probability distributions (on the same measure space) are s.t they have p.d.fs and the $L^1$ distance between the p.d.f.s is large, then is there a choice of a ``nice" metric $d_{\rm ...
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Bounding the total variation metric between Gaussian mixtures
Let $\mathcal{P}(\mathbb{R}^d)$ the space of probability measures on $\mathbb{R}^d$ with total variation metric $\delta$, fix $k \in \mathbb{N}$, and let $\mathcal{P}'\subset \mathcal{P}(\mathbb{R}^d)$...
- 5,405
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A map similar to a foot map on a cap of a strictly convex surface
Consider a strictly convex surface $\Sigma$ in $\mathbb{R}^3$ homeomorphic to a sphere. When $p$ is a point not in a convex hull of $\Sigma$, then $\Sigma'$ is the boundary of convex hull of $p$ and $\...
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Can every quadrangle whose corners are in little squares be more square-like when we allow corners to move in the little squares [closed]
Let $f: P^4 \to \mathbb R^+$, $(A,B,C,D)\mapsto (AB-BC)^2+(BC-CD)^2+(CD-DA)^2+(AC-BD)^2$.
Where $AB$ is the distance betwin $A$ and $B$.
Let $A$, $B$, $C$, $D$ be four points of the plane $P$, ...
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Represent the metric of $n$ points in the Euclidean space [closed]
I'm trying to understand better the following sentence (see link to the reference at the end):
To represent the metric of $n$ points in the Euclidean space, one
clearly needs no more than $n−1$ ...
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Compatibility of the Hausdorff measure with short exact sequences in normed spaces
Let $(E,\|.\|)$ be a finite dimensional normed space and take $F\subset E$ a
subpace, so that we have the canonical short exact sequence
$0\rightarrow F\rightarrow^\iota E\rightarrow^\pi E/F\...
- 313
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179
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Curvature of Lie groups
Can a Lie group be of negative sectional curvature?
In fact every Lie group contains an entire curve which means that a Lie group does not admit a complete metric with negative sectional curvature
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348
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Sectional curvature in complex manifold
Let $(X, \omega)$ be a Hermitian manifold .Say that the sectional curvature of X is negative is the same to say that the sectional curvature of the Hermitian metric $\omega$ is negative, otherwise, ...
- 11
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79
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A Lipschitz limit of Riemannian metrics with curvature in $[-1,1]$
Let $(M,g)$ be a compact manifold with a metric $g$ (not necessarily Riemannian one). Suppose that $(M,g)$ is a Lipschitz limit of a sequence of Riemannian manifolds $(M_n,g_n)$ with the sectional ...
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91
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Gaussian width and restricted isometry
It is known that, for an $m$ dimensional space and an $n\times m$ dimensional random matrix $U$ whose entries are iid Gaussian, then $\|I-(1/n)U^TU\|$ is bounded by $\sqrt{m/n}$ when $n>m$.
If a ...
- 11
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How to project a matrix to a unitary matrix?
Given a nonzero vector $v \in \mathbb{R}^n$, we all know that it's projection onto the unit $\ell_2$ ball is just $\frac{v}{\|v\|}$. Let $X$ be some nonzero $n \times n$ matrix. What is the projection ...
- 1,703
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361
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Geometric interpretation of metric [closed]
For metric in $\mathbb R^2$ (polar coordinates) Pythagorean triangle elements can be visualized in a differential form as seen at top yellow triangle. This happens for metric:
$$ ds^2= dr^2 +(r d \...
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A Hölder version of the Johnson-Lindenstrauss Lemma on essentially bounded functions
Does there exist a Hölder (not necessarily linear) projection from $L^{\infty}(\mathbb{R}^d)$ to any finite-dimensional linear subspace? This is known when $L^{\infty}(\mathbb{R}^d)$ is replaced by a ...
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