Questions tagged [mg.metric-geometry]

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

Filter by
Sorted by
Tagged with
0 votes
0 answers
67 views

CAT(k) spaces and law of cosines

Let $M^n_{\kappa}$ be a model space with curvature $\kappa>0$. The sides are $a,b,c$, and the angle at the vertex opposite to the side of $c$ is $\gamma$. $c$ is known. Consider the corresponding ...
Nelson's user avatar
  • 1
4 votes
0 answers
94 views

Does the permutohedron satisfy any minimal distortion property for graph metric vs Euclidean distance?

We can look on the permutohedron as a kind of "embedding" of the Cayley graph of $S_n$ to the Euclidean space. (That Cayley graph is constructed by the standard generators, i.e. ...
Alexander Chervov's user avatar
5 votes
1 answer
115 views

Finding the point within a convex n-gon that maximizes the least angle subtended there by an edge of the n-gon

For any point P in the interior of a convex polygon, the sum of the angles subtended by the edges of the polygon is obviously 2π. Given a convex polygon, how does one algorithmically find the point (...
Nandakumar R's user avatar
  • 5,473
2 votes
0 answers
63 views

What Cayley graphs arise as nodes+edges from "nice" polytopes and when are these polytopes convex?

The Permutohedron is a remarkable convex polytope in $R^n$, such that its nodes are indexed by permutations and edges correspond to the Cayley graph of $S_n$ with respect to the standard generators, i....
Alexander Chervov's user avatar
3 votes
1 answer
101 views

Concentration of measure on spheres with respect to a unitary of trace approximately zero

Cross-posted from MSE, where it hasn’t received any answer yet: This question arose out of my attempt to understand how a unitary of trace approximately zero acts on the unit sphere of a $n$-...
David Gao's user avatar
  • 1,262
6 votes
1 answer
152 views

Coarse embeddings and Gromov products in (Gromov) hyperbolic spaces

I am new into geometric group theory and I have recently started reading the book "Sur les Groupes Hyperboliques d’après Mikhael Gromov" by Ghys and de la Harpe. The following inequality ...
Steve's user avatar
  • 61
2 votes
0 answers
52 views

Limiting distribution of separated points in a unit square

Let $n$ and $r$ be fixed, and consider the following process, with $S=\emptyset$ to start: For $i\in\{1,\dots,n\}$: Sample a random point $X$ in the unit square. If $X$ is a distance at least $r$ ...
Tom Solberg's user avatar
  • 3,929
1 vote
1 answer
69 views

Simple convergence of convex compact set implies Hausdorff convergence

I am wondering about the following : In $\mathbb{R}^n$, suppose you are given compact convex bodies $\left\{ C_k : k \geq 1 \right\}$ and $C$, such that for every $x \in \mathbb{R}^n$ $$ \mathbb{1}_{...
Anthony's user avatar
  • 41
7 votes
1 answer
330 views

Proving the inequality involving Hausdorff distance and Wasserstein infinity distance

Prove the inequality $$d_{H}(\mathrm{spt}(\mu),\mathrm{spt}(\nu))\leq W_{\infty}(\mu,\nu)$$ where $d_H$ denotes the Hausdorff distance between the supports of the measures $\mu$ and $\nu$, and $W_\...
Luna Belle's user avatar
1 vote
1 answer
108 views

Does a Riemannian submersion map horizontal geodesics to geodesics, and a relevant question?

I asked this question on MSE, but I didn't receive a response yet, so I'm asking here. Apologies if the question is not exactly a research level question, but I'm having some trouble in figuring them ...
Learning math's user avatar
5 votes
1 answer
242 views

Quotients in categories of metric spaces

There are several categories whose objects are metric (or pseudo-metric) spaces. Natural choices of morphisms are continuous, uniformly continuous, Lipschitz or short (= non-expansive or contractive) ...
Jochen Wengenroth's user avatar
21 votes
1 answer
1k views

Does greedy circle packing exhaust the measure of every bounded open set in the plane?

The greedy circle packing of a bounded region in the plane is the result of placing at each stage the largest possible disk into the region that remains uncovered. The greedy circle packing of a ...
Joel David Hamkins's user avatar
9 votes
1 answer
328 views

Is the group of translations of an affine plane always commutative?

$\DeclareMathOperator\Dil{Dil}\DeclareMathOperator\Trans{Trans}\DeclareMathOperator\Col{Col}$An affine plane is a set of points $X$ endowed with a family $\mathcal L$ of subsets of $X$, called lines, ...
Taras Banakh's user avatar
  • 40.8k
1 vote
0 answers
39 views

Polyhedra inscribed in a sphere with mutually non-congruent, equal area faces

Two constrained versions of the main question given in this post: Polyhedrons with mutually non-congruent faces, all of equal area. An earlier post that could be related: Cutting a spherical surface ...
Nandakumar R's user avatar
  • 5,473
2 votes
0 answers
233 views

Convergence of metric and eigenvalues on a tubular neighbourhood

Background: Consider the sphere $S^2$ with the round metric $g$ and let $\gamma$ be one half of a great circle of length $\pi.$ Let $T_\epsilon$ denote a geodesically convex tube around $\gamma$ of ...
Student's user avatar
  • 653
0 votes
0 answers
31 views

Calculate the intersection volume of two spherical caps on the same sphere [migrated]

My question is the same as this question except that I am looking for the intersection volume of two caps instead of the area. Given a sphere with radius $R$ that has two spherical caps with base ...
Danny's user avatar
  • 1
4 votes
0 answers
107 views

Curiosity about "conditional trig identities"

Perhaps this should be cross-posted on Math Stackexchange, but it came up in the context of some research mathematics (quaternion orders, etc.) In this context, I have three angles $\alpha, \beta, \...
Marty's user avatar
  • 13.1k
11 votes
0 answers
308 views

Is a convex polyhedron determined by its edge lengths and angular defects?

Let's consider 3-dimensional convex polyhedra $P\subset\Bbb R^3$. The angular defect at a vertex $v$ is $2\pi$ minus the sum of the interior angles of the incident faces at $v$. Question: Is a ...
M. Winter's user avatar
  • 12.5k
-2 votes
1 answer
60 views

Inner Products of Elements in Spherical Cap [closed]

I am interested in understanding what is the lowerbound on the inner products of two elements of a sphere. Based on my intuition in dimension 2, I come up with the following conjecture. I appreciate ...
MMH's user avatar
  • 129
2 votes
0 answers
55 views

For riemannian manifolds, how close can a mapping from atlas be to an isometry?

Let $(M, g)$ be an $n$-dimensional $C^k$ (or $C^\infty$) Riemannian manifold. On $M$ we can define metric $d_g$ as the infimum of lengths of curves that connect given two points. Fix $x \in M$ and $r&...
Kacper Kurowski's user avatar
9 votes
0 answers
212 views

Existence of $1$-separated and $(1-\varepsilon)$-dense set in metric spaces

Is it know which metric spaces $M$ do have the following property: there is $\varepsilon>0$ and a maximal $1$-separated set which is $(1-\varepsilon)$-dense? In other words, when does at set $S\...
Christian's user avatar
  • 779
3 votes
0 answers
84 views

Are 1-Wasserstein and 2-Wasserstein distances between multivariate normal distributions equivalent?

The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by $$W^p_p(\nu_{1},\nu_{2})=\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,dy)$...
Vladimir Zolotov's user avatar
10 votes
1 answer
259 views

Do triple-linked graphs exist?

Lets say that a finite simple graph $G$ is (intrinsically) fully triple-linked if for each embedding of $G$ into $\Bbb R^3$ we can find three disjoint cycles $C_1,C_2,C_3\subset G$ whose embeddings ...
M. Winter's user avatar
  • 12.5k
4 votes
1 answer
145 views

Reference request: Fréchet embedding

Given a separable metric space $(X,d)$, we have an isometric embedding $\iota:X\to\ell^\infty$ given by taking $(x_n) _{n \ge 0}$ to be the countable dense subset and sending $\iota(x)_n=(d(x,x_n) - d(...
Gesh's user avatar
  • 105
1 vote
1 answer
117 views

Connectedness of fibers of almost Riemannian submersions

EDIT: Let $M,N$ be compact connected smooth Riemannian manifolds. Let us assume that $N$ is closed, while $M$ might have a geodesically convex boundary. Given $f\colon M\to N$ be an $\varepsilon$-...
asv's user avatar
  • 21.1k
1 vote
0 answers
62 views

Algorithm to generate configurations with kissing number 12

That the kissing number of a sphere in dimension 3 is 12 is well known. However, it is also known that there is a lot of empty space between the 12 spheres. I deduce (am I wrong?) that there are many ...
GRquanti's user avatar
  • 111
6 votes
0 answers
155 views

Does there exist a plane curve such that it has the heart curve as catacaustic?

Given a curve $C$ and a fixed point $L$ (the light source), the catacaustic of $C$ with respect to $L$ is the envelope of light rays coming from $L$ and reflected from the curve $C$. The catacaustic ...
zemora's user avatar
  • 545
5 votes
0 answers
129 views

Given a collection of vectors $x_1,\ldots,x_k$, which inner products $\langle x_i,x_j\rangle$ are needed to uniquely determine all inner products

Given a collection of vectors $x_1,\ldots,x_k$, which inner products $\langle x_i,x_j\rangle$ need to be known to uniquely determine all inner products? I'll begin with the specific case I am ...
RandomTensor's user avatar
1 vote
2 answers
263 views

Does $\mathbb Z^n$ contain $A_n$?

Are there any positive integer $n > 3$ such that the root lattice $A_n$ is contained in $\mathbb Z^n$?
WKC's user avatar
  • 646
0 votes
0 answers
53 views

Seek a partition of $\mathbb R^d$

Let $c>0$ be given. I look for $n\ge 1$ and a collection of closed subsets $(F_i: 1\le i\le n)$ such that $$\bigcup_{1\le i\le n} {\rm int}(F_i)= \mathbb R^d,$$ and for every $x\in \mathbb R^d$, ...
GJC20's user avatar
  • 1,220
1 vote
1 answer
87 views

To place copies of a planar convex region such that number of 'contacts' among them is maximized

A contact between two planar convex regions obviously happens either along a line segment or at a single point. Question: Given a planar convex region $C$ and a number $n$, we need to lay out $n$ ...
Nandakumar R's user avatar
  • 5,473
-2 votes
1 answer
122 views

Interpretation and validity of modified Heisenberg uncertainty principle in a metric context? [closed]

Considering the Heisenberg uncertainty principle, which states $\Delta x \cdot \Delta p \geq h$, I've explored a modified version by computing $(\Delta x + 1)(\Delta p + 1) \geq \Delta x \cdot \Delta ...
mathoverflowUser's user avatar
1 vote
1 answer
84 views

When "$(\varepsilon,\delta)$-geodesic" cannot be a loop?

EDIT: Let $M$ be a smooth compact Riemannian manifold. Let $\varepsilon,\delta>0$. I call a smooth curve $\gamma\colon [a,b]\to M$ an $(\varepsilon,\delta)$-geodesic if for any $t_1<t_2<t_1+\...
asv's user avatar
  • 21.1k
3 votes
1 answer
193 views

"Almost geodesics" in Riemannian manifolds which cannot be loops

Let $M,N$ be smooth Riemannian manifolds of the same dimension. Let $0<\varepsilon<\frac{inj(N)}{100}$. Let $f\colon M\to N$ be a smooth map such that for any $x\in M$ and any $v\in T_xM$ one ...
asv's user avatar
  • 21.1k
6 votes
0 answers
178 views

When is a distance space dominated by a metric space?

A distance space is a pair $(X,d)$ where $X$ is a set and $d:X \times X \rightarrow \mathbb{R}$ is a symmetric, non-negative map such that $d(x,x)=0$ for all $x \in X$. These are sometimes called semi-...
David Bryant's user avatar
1 vote
1 answer
57 views

To maximize the volume of the polyhedron resulting from perimeter-halvings of a convex polygonal region

We add one more bit to Forming paper bags that can 'trap' 3D regions of max surface area (note: some possibly open related questions are also in the comments following the answer to above ...
Nandakumar R's user avatar
  • 5,473
3 votes
1 answer
75 views

Embedding a countably infinite metric space in $\ell^2(\mathbb Z_+)$

Suppose $(X,d)$ is a countably infinite set endowed with a metric $d$ that satisfies the following condition: Every finite subset of $X$ with the induced metric is isometric to a subset of some ...
Daniel Asimov's user avatar
2 votes
0 answers
76 views

Nested convex hulls in Hadamard manifold

Let $F$ be a finite set in a Hadamard manifold $H$, and $W\supset F$ is its neighborhood. Is it true that the closure of the convex hull of $F$ lies in the interior of the convex hull of $W$? ...
Anton Petrunin's user avatar
0 votes
0 answers
38 views

What is the dual of a hyperbolic configuration of points?

Let $C_n$ denote the configuration space of $n$ distinct points in hyperbolic $3$-space $\mathcal{H}$. If $\mathbf{x} := (\mathbf{x}_1, \dots, \mathbf{x}_n) \in C_n$, where $\mathbf{x}_i \in \mathcal{...
Malkoun's user avatar
  • 4,991
1 vote
0 answers
90 views

A claim on the largest area circular segment that can be drawn inside a planar convex region

This post adds a little to To find the longest circular arc that can lie inside a given convex polygon A circular segment is formed by a chord of a circle and the line segment connecting its endpoints....
Nandakumar R's user avatar
  • 5,473
3 votes
0 answers
91 views

Sampling uniformly from the convex cone

Let $n$ vectors of dimension $d$ (e.g., $n = 100$, $d = 10000$), each with infinity norm of $1$, be given. The conic combination of those $n$ vectors generates a convex cone. How to uniformly sample ...
mathhamcs's user avatar
3 votes
0 answers
150 views

What are the measure of the volume and boundary (and other quermaß measures) of the positive semidefinite matrices?

Let $E$ be the real vector space of $n\times n$ real symmetric (resp. complex Hermitian) matrices, and $E_1$ those with trace $1$. Endow $E$ with the bilinear (resp. sesquilinear) form given by $(P,Q)...
Gro-Tsen's user avatar
  • 29.9k
1 vote
2 answers
194 views

A triangle comparison in CAT(0) spaces

Let $pxy$ be a triangle in a CAT(0) space $X$, and $p' x' y'$ be a triangle in $\mathbf{R}^2$ such that the lengths $|px|=|p'x'|$, $|py|=|p'y'|$ and the angle $\angle(xpy)=\angle(x'p'y')$. Let $z\in ...
Mohammad Ghomi's user avatar
0 votes
0 answers
46 views

Which planar convex region with specified area and perimeter maximizes/minimizes Moment of Inertia?

By moment of inertia of a planar convex region C, here we mean its moment of inertia about an axis passing through the center of mass of C and perpendicular to the plane of C. Question: For specified ...
Nandakumar R's user avatar
  • 5,473
2 votes
1 answer
165 views

Forming paper bags that can 'trap' 3D regions of max surface area

An existence question based on 'Trapping' 3D regions with sheets of paper. Given a sheet of paper S that is a planar convex region, one tries to form a 'closed bag' that contains a connected ...
Nandakumar R's user avatar
  • 5,473
4 votes
1 answer
128 views

Characterization of convexity by connectedness of hyperplane sections

Let $S$ be a smooth closed connected embedded hypersurface in $\mathbb R^n$. Is it true that $S$ is convex, i.e. is a boundary of a convex set, if and only if any section of $S$ by a hyperplane is ...
Dmitrii Korshunov's user avatar
2 votes
0 answers
46 views

Instances of c-concavity outside of optimal transport?

Let $X$ and $Y$ be metric spaces, and let $c:X\times Y\rightarrow \mathbb{R}$ be a nonnegative function which we refer to as a cost. For any $\phi:X\rightarrow \mathbb{R}$ and $\psi:Y\rightarrow \...
Brendan Mallery's user avatar
2 votes
1 answer
150 views

Osculating sphere at point of maximal curvature lies to one side

I'm looking for the higher dimensional version of this post, which says that given a curve $\gamma \subseteq \mathbb{R}^2$, the osculating circle will lie to one side of the curve at points of maximal ...
JMK's user avatar
  • 301
0 votes
0 answers
69 views

On 'Width Equalizers' of planar convex regions

Definitions: The least width of a 2D convex region C is the least distance between any pair of parallel lines that both touch the boundary of C (in what follows, we refer to this quantity as simply '...
Nandakumar R's user avatar
  • 5,473
3 votes
1 answer
147 views

Equivalent definition for Skorokhod metric

I have a question about the Skorokod distance on the space $\mathcal{D}([0,1],\mathbb{R})$: $$ d(X,Y):= \inf_{\lambda \in \Lambda}\left( \sup_{t\in [0,1]}|t-\lambda(t)|\vee \sup_{t\in [0,1]}|X(t)-Y(\...
user1598's user avatar
  • 177

1
2 3 4 5
79