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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Are there two tetrahedrons with the same volume that share their opposite edge lengths and arent the same or a different chirality of the same? [closed]
I have been coming up with an efficient way to decide if two tetrahedrons are similar. I believe that it is enough for a computer to check for the ordered by length list of pairs of opposite edges on ...
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174
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Need help maximizing distances to nearest neighbor in a cylinder
I have a cylinder and I want to maximize the number of points in the cylinder such that the distances to the nearest neighbors are maximally spaced. How do I find out how many points I can have so ...
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Name for a uniform local boundedness property of a function
I am working with a function $f : \mathbb{R}^N \to \mathbb{R}$ having the property that for every $R > 0$, there exists $M > 0$ such that if $x, y \in \mathbb{R}^N$ and $\vert x - y \vert \le R$,...
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Rigidity in a $CAT(-1)$ space
Summary: How to proove that a reunion of triangles in a
$CAT(-1)$ space is isometric to the reunion of coresponding
comparisons triangles ?
Context and notations:
Le $X$ be a $CAT(-1)$ metric space.
...
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Does a continuous function have a continuous integral function in a discrete dynamical system?
Let $X$ be a compact manifold (or the closure of a Euclidean domain if that helps significantly) and $T\colon X\to X$ a homeomorphism.
Let us say that a function $v\colon X\to\mathbb R$ is the ...
- 8,086
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Existence or otherwise of a set of "sufficiently intricate" open cells
In my question Existence or otherwise of a set of "sufficiently intricate" open sets, I asked about whether it is possible to partition Lebesgue-almost all of $\mathbb{R}^d$ into a finite ...
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Is the map $\exp_x(\nabla_x \sum_{i=1}^m d^2(x_i,x))$ Lipschitz?
The last question is too general, this is a modification.
Let $M$ be an $n$ dimensional Riemannian manifold. Fix $m$ points $x_1,...,x_m$. Suppose $y$ is not in the cut locus of $x_i$ for $1 \...
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An optimality condition for the corners of convex polytopes?
Let $H,H'\subset\mathbb{R}^m$ be two hyperplanes with unit normal-vector,
and let $P\subset\mathbb{R^m}$ be a convex polytope (defined via its corners $v_0, ... , v_n$, where $n\ge m$).
Let's ...
- 13.2k
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Convergence in the Wasserstein metric and the square root function
Let $f$ be a smooth probability distribution on the unit square $S$ such that $f(x)>0$ on $S$. Let $\{g_i\}$ be a sequence of smooth probability distributions such that $g_i(x)>0$ on $S$ as ...
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Does the singular cohomology for a metric space of finite topological dimension vanish in high dimensions?
It is known that by applying the universal coefficient theorem, the singular cohomology of closed manifold with coefficient $\mathbb{Z}_2$ vanishes in high dimensions. But for a metric space $M$ with ...
- 285
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Generalization of the triple tangent identity
It is well known that if $x + y + z = \pi$ then $$\tan x \times \tan y \times \tan z = \tan x+ \tan +\tan z.$$
I came across the following generalization of this equality:
$$\sqrt{1-k^2} {\rm sc}(...
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337
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Estimating the volume of a union of balls
Let $\{ B_i \}_{i=1}^n$ be a set of $n$ ball in the unit cube $C$ of dimension $d$.
If I want to estimate
$$
\frac{ \lambda \left( \cup B_i \right) }{\lambda\left( C \right) }, \tag{1}
$$
where $\...
user24451
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Probabilistic Johnson-Lindenstrauss Lemma for arbitrary points
Consider the following standard formulation of the Johnson-Lindenstrauss lemma:
Lemma (JL).
For any $0<\epsilon < 1$ and any integer $n$, let $k$ be a positive integer such that $k\geq C\...
- 424
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Length of the transversal for surfaces with cusps
In Peter Buser's Geometry and Spectra of Compact Riemann Surfaces he shows that the length of the transverse curve to a geodesic in a pants decomposition on a compact hyperbolic surface has length a ...
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909
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Heat flow $P_tf \to f$ in $W^{1,2}$ for $f \in W^{1,2}$?
$\varepsilon:L^2(X,m) \to [0,\infty]$ is a strongly local, symmetric Dirichlet form generating a Markov semigroup $(P_t)_{t\ge0}$ in $L^2(X,m)$. Let $D(\varepsilon)=\{f\in L^2(X,m):\varepsilon(f)<\...
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Embedding of Two Objects Into Higher Dimensions With Their Sum
Given two vector sets, $\vec x_i$ and $\vec y_i$ (for $i$=1,2,...N, but the dimensionality of each vector can be more than N), let their sum set be $\vec z_i = \vec x_i + \vec y_i$. It's easy to ...
- 1,547
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Construction of an integral point set given the set of distances, its minimal description to get a measure of its complexity and its unique identifier
Given a set of distances between every pair of points of an integral point set $P$ of $n$ points; say $D = \{{d_i}\}$.
Q1. What is the least time complexity
possible/known for recreating the
...
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Inferring the properties of a visibility blocker tangential to a point-like light source
Imagine there's a point-like particle undergoing radioactive decay at some position $(0,0,0)$ in Euclidean $3$-space. We encapsulate this particle with a spherical detector for the decay products it ...
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127
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3-dim 1-connected Alexandrov manifold with curvature $\ge 0$ Heomomorphic to sphere?
For Alexandrov manifold in the title we mean 3-dim Alexandrov apace which is also a topological. manifold.
Shioya-Yamaguchi posted a conjecture on their paper "Collapsing 3-manifold with lower ...
- 283
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290
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Examples on small cut radius of totally convex set in non-negatively curved manifold
Suppose $M^n$ is an open complete nonnegatively curved Riemannian manifold. In Cheeger-Gromoll's proof of the soul theorem. They need an estimate on the cut radius of a totally convex set $C$. By a ...
- 283
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Orbits of Product Lie Groups Action
Hi to all,
Let $G$ be a Lie group of linear isometries of $\mathbb{R}^n_{\nu}$ ($\mathbb{R}^n_{\nu}$ is the semi-Euclidean space) and $G_1$ ,$G_2$ two Lie subgroups of $G$. Let $G_1 \times G_2$ as ...
user30435
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618
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Why $O(4n,\mathbb{C})$ (orthogonal group) acts transitively on the space of maximal isotropics of $V\bigotimes \mathbb{C}$ ?
We say $L< (V\oplus V^{*})\bigotimes \mathbb{C}$ is isotropic when $< X,Y>=0$ for all $X,Y\in L$
Why $O(4n,\mathbb{C})$ (orthogonal group) acts transitively on the space of maximal ...
user21574
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1
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516
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Properties of the minimum circumscribing circle for points sequentially placed within a circle of radius $r$ of previously placed points on a 2D plane
Imagine I perform the following procedure:
[1] At time point $t_1$, I place a single point on a two-dimensional plane at the coordinate $(x, y) = (0, 0)$.
[2] At time point $t_2$, I center a ...
- 197
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111
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Log-nonexpansive functions: terminology and references
During my recent work in the optimization of positive valued functions, the following class of functions proved to be exceptionally important.
(Defn.). Let $h: (0,\infty) \to (0,\infty)$ be ...
- 28.6k
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421
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Triangles with Congruent Corresponding Sides that Cannot fold into a Tetrahedron
I've been trying to find, without much success, 4 triangles whose corresponding sides are congruent that cannot be folded into a tetrahedron.
Anyone has any clue how to approach this problem?
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342
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Half-space comparison of perimeter
Claim: suppose that $E$ is a set of finite perimeter, and $H$ is a half space. Then $P(F\cap H)\le P(F)$. In words: restricting a Caccioppoli set to a half-space will not increase the perimeter.
My ...
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390
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Isocontours of depth and magnitude of gradient
We are interested in characterizing a 2D surface $z(x,y)$, where $(x,y)$ is the regular 2D Cartesian grid. Let $\nabla z = (z_x, z_y)$ denote the gradient. The surface is a "general" one, that is, ...
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What are the topological properties of the metric space retained (inherited) for its completion
Let $(X,d)$ be a metric space and $(\bar{X},\bar{d})$ its completion.
There is a list of topological properties
Wikipedia - Topological property
Does anybody know list which of them are retained (...
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419
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Is the direction of the longest line of a polytope unique?
The question pertains to a polytope that is generated by the intersection of an affine subspace with a hypercube in $p$ dimensions.
The affine subspace is given by:
$X \mbox{ u} = y$
where
$u$ &...
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On $N$-partition of some common subsets $\Omega\subset\mathbb R^d$
Let $\Omega\subset\mathbb R^d$ be compact and convex, and denote by $\ell$ the normalised Lebesgue measure such that $\ell(\Omega)=1$. Let $N$ be an arbitrary but fixed integer.
In this post we set $d=...
- 1,399
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Quasi-geodesics in Alexandrov spaces
I am trying to understand the notion of quasi-geodesic in Alexandrov space with curvature bounded below following the Perelman-Petrunin paper. I have two questions:
Is it true that the shortest ...
- 21.8k
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Metric entropy of an ellipsoid
Let $B^d_2$ denote the unit ball of $\ell_2^d$ and let $T$ be an invertible linear map.
Consider the function
$$
H(T) := \log M(TB_2^d, B_2^d),
$$
which is the packing entropy for $TB_2^d$ by $B_2^d$....
- 420
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Cut locus of linear isometric action quotients
Given a compact group $G\leq \operatorname{O}(d)$ of linear isometries on $\mathbb R^d$, equip its quotient $\mathbb R^d/G$ with the canonical orbital metric.
I am curious about the following. Is ...
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Collapse of Moebius bands with bounded below Gauss curvature and convex boundary
Let $\{M_i\}_{i=1}^\infty$ be a sequence of (compact) Moebius bands with Riemannian metrics with Gauss curvature at least $-1$ and such that the boundaries are geodesically convex (equivalently, the ...
- 21.8k
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Obtaining the geodesic extension property by embedding in a larger space
Suppose $(X,d)$ is a Hadamard space. By considering basic examples like a compact interval in $\mathbb{R}$ or a closed unit ball in Hilbert space, $X$ need not have the geodesic extension property (...
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Low dimensional symmetric Euclidean immersions of the Klein quartic
I was wondering what the lowest Euclidean dimension the Klein quartic has a faithful symmetric immersion, that is an immersion such that all of its automorphisms* are distinct symmetries of the ...
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Is the circumcenter Lipschitz on large convex sets in hyperbolic space?
Given a uniquely geodesic metric space $X$, let $\mathcal K(X)$ denote the metric space of compact, convex subsets of $X$ equipped with the Hausdorff distance. Given $K \in \mathcal K(X)$, let $c(K)$ ...
- 498
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A certain circle formed by perpendiculars
If six points are chosen, two points on each side of a triangle, such that they have the same ratio of distances to vertices, then the perpendicular lines through those points meet at six concyclic ...
- 1,109
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Contraction and consensus on Hadamard manifolds
Let $\mathcal M$ be a Hadamard manifold and $\{x_i\}_{i=1}^n\subseteq{\cal M}$ be $n$ points. Define $\{y_i\}_{i=1}^n$ as the weighted Fréchet means:
$$
y_i=\arg\min_{y\in\mathcal M}\sum_jw_{ij}d^2(y,...
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Seeking Help with Classifying Polygons: Waterholes and Airpockets in 2D Space
I am currently in the process of writing software and have encountered a mathematical problem. Perhaps there are some experts here who are familiar with this. It involves the classification of ...
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Newton-Kantorovich: theorem geometric
This post is cross-posted from Math StackExchange where I did not receive any response after 5 days. I guess this question might be targeted more towards research level mathematics, so I decided that ...
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Name of the perspector of the orthic triangle and excentral triangle
The orthic triangle and tangential triangles of a given triangle are in perspective. What's the official kimberling center associated with this perspector?
- 1,109
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A circle is inscribed in a triangle, with three other circles in the corner regions. The radii are integers. Possible values of the largest radius?
Originally posted at MSE.
A circle with integer radius $R$ is inscribed in a triangle. Three other circles with integer radii $a,b,c$ are each tangent to the large circle and two sides of the ...
- 3,527
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Conjecture on the increasing efficiency of the shortest minimum-link polygonal chains covering any grids of the form $\{0,1,2\}^k$ as $k$ grows
From the well-known Nine dots problem, we know that we need a polygonal chain with at least $4$ edges to connect the $9$ points of the planar grid $G_{3,2}:=\{\{0, 1, 2\} \times \{0, 1, 2\}\} \subset \...
- 1,451
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On a pair of solids with both corresponding maximal planar sections and shadows having equal area
This post pulls together Are two convex solids with all corresponding shadows equal in area congruent? and
What can be said about 2 convex solids with corresponding maximal planar sections having ...
- 5,979
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What can be said about 2 convex solids with corresponding maximal planar sections having equal area?
This post follows Are two convex solids with all corresponding shadows equal in area congruent?
Every convex 3D body has planar sections with normals in any given direction. We consider the maximum ...
- 5,979
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Is the formula known? and can we generalized for higher dimensions?
In this configuration as follows, we have a nice formula:
$$\cos(\varphi)=\frac{OF.OE+OB.OC}{OF.OB+OE.OC}$$
Is the formula known? and can we generalized for higher dimensions?
- 1,776
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Special rectangle and its existence in non-Euclidean geometries
My questions is motivated by Folding the Hyperbolic Crane article which presents non-Euclidean paper for origami and the existence of a special rectangle on Euclidean paper.
Actually, there exists a ...
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42
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Genaralizing the metric expression present in the quadrilateral inequality
Let $(X, d)$ be a metric space. In Sato - An alternative proof of Berg and Nikolaev’s characterization of CAT(0)-spaces via quadrilateral inequality it is stated that if $X$ is a geodesic space, then ...
- 820
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Good orbifold and Ricci flow with Dirichlet boundary conditions on $\Sigma$
An orbifold $\mathcal O$ is a metrizable topological space equipped with an atlas modeled on $\Bbb R^n/\Gamma, \Gamma<O(n)$ finite. Let $\Sigma$ be the singular locus i.e. points modeled on $\...
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