# Questions tagged [mg.metric-geometry]

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

3,856
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### Conformal Killing vector fields on manifolds that are not asymptotically flat

Let $M = [1,\infty) \times S^2$.
Equip $M$ with the metric $g = dr^2 + r^2 (\gamma + h)$ where $\gamma$ is a metric on $S^2$ and $h$ is a $(0,2)$ tensor on $M$ that satisfies
$$h = O(1/r),\quad \...

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2
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### Conditions on a parametric curve so that its normal plane covers R^n

I am working on a control theory problem that either just caught me on a blind spot or is beyond me. I guess it's not a new question, but I couldn't find anything about it.
Let $p(s), s\in \mathbb{R}, ...

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### Variants of the Bonk-Schramm embedding

Recently I heard about the following embedding theorem of Bonk and Schramm: every Gromov hyperbolic geodesic metric space with "bounded growth" is roughly similar to a convex subset of $\...

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### What is the minimum and the maximum perimeter of a triangle with area $x$ that can be inscribed in a circle?

This question was posted in MSE but is still open hence posting in MO.
The area of the largest triangle that can be inscribed in a circle of raidus $1$ is $\displaystyle \frac{3 \sqrt{3}}{4}$ for a ...

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### Groups acting non-properly cocompactly on hyperbolic spaces

A group $G$ is hyperbolic if it admits a geometric (the action is proper and co-bounded) action on a geodesic hyperbolic metric space. Also, the definition can be given as follows, a group $G$ ...

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### Fitting a simplex to set of almost orthogonal vectors

I am trying to solve the following question, that I guess (hope?) has been solved before but I couldn't find any reference.
Let $S$ be a set of $d$ unit vectors in a $d$-dimensional Euclidean space ...

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### Injective hulls of metric spaces

In the context of large scale geometry and geometric group theory, I have recently come across the concept of injective hulls of metric spaces. For a metric space $X$, let $\text{In}(X)$ be the set of ...

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### Conjecture: Given any five points, we can always draw a pair of non-intersecting circles whose diameter endpoints are four of those points

The following question resisted attacks at Math SE, so I thought I would try posting it here.
Is the following conjecture true or false:
Given any five coplanar points, we can always draw at least ...

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### Moser iteration epsilon-regularity for non-linear system in general dimension

I am attempting to prove the following result in general dimension $n$. Given $(M^n,g)$ a Riemannian manifold with $\mathrm{Ric}_g \geq -(n-1)$ and $\mathrm{Vol}_g(B_1(x)) \geq v > 0$ for all $x \...

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### Average distance between points of lower dimensional simplices in $\mathbb R^n$

Notation: By a simplex, we mean the convex hull of a finite set of distinct points in $\mathbb R^n$, which are called the vertices of the simplex. $\mathcal H^n$ will denote the $n$-dimensional ...

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### A question about the existence of surjective contractions

A few years ago I was doing some research in origami, and was motivated to as the following questions:
Consider $\mathbb{R}^2$ with the Euclidean metric and Lebesgue measure. Does there exist a ...

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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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### How do we calculate the gradient of this function defined using the Riemannian logarithm on a Riemannian manifold?

We consider the following function $\psi$ on an open subset $V\subset M,$ a Riemannian manifold of dimension $m,$ so that $\exp_p:U\to V$ is a diffeomorphism with its inverse $\log_p: V\to U$. Let $v\...

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### Convex hull of 3 points in Cartan-Hadamard manifolds

Can the convex hull of $3$ points in a Cartan-Hadamard manifold be smooth?
A Cartan-Hadamard manifold $M$ is a complete simply connected manifold with nonpositive curvature (so it includes the ...

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2
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### Convergence of metric spaces of increasing dimension

Given two metric spaces we can define the Gromov-Hausdorff (GH) distance. There are compactness results stating that a sequence of manifolds of a fixed dimension, with a uniform lower Ricci bound and ...

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### To find the convex planar region minimizing diameter when area and perimeter are given

The basic question is to find that planar convex region for which diameter is a minimum when area and perimeter are specified.
A partial answer is given here: http://nandacumar.blogspot.com/2012/11/...

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### Will a unit disk be completely covered by randomly placed disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ with probability $1$?

On a "bottom" disk of area $\pi$, we place "top" disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ such that the centre of each top disk is an independent uniformly random ...

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### Is every finite metric space representable in a pseudo-Euclidean space?

Let $X$ be a finite set with a (true) metric $d$ and $|X| = n$. Does there exist a set $Y$ of $n$ points in $R^n$ with a pseudo-Riemannian metric with signature $(n - k, k, 0)$ for some integer $k$ ...

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### Question about $n$ random points in a regular polygon, and a limiting probability

Suppose we choose $n$ uniformly random points in a disk, then draw the smallest circle that encloses all of those points. There is evidence suggesting that the probability that the enclosing circle is ...

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### Amalgamated product acting on CAT(0) cube complex

I was reading the following result from the book Metric spaces of non-positive curvature by Bridson and Haefliger.
Result:
Let $F_0,F_1$ and $H$ be groups acting properly
by isometries on complete $...

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### Intuition behind right-inverse of map from Johnson-Lindenstrauss Lemma

The Johnson–Lindenstrauss lemma states that for every $n$-point subset $X$ of $\mathbb{R}^d$ and each $0<\varepsilon\le 1$, there is a linear map $f:\mathbb{R}^d\to\mathbb{R}^{O(\log(n)/\varepsilon^...

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### A gerrymandering problem - can you always turn a tie into a landslide victory?

Note: Here we use $|A|$ to denote the Lebesgue measure of a measurable subset $A$ of $\mathbb R^2$.
Your party is running for election! In your country, voters are approximately uniformly distributed. ...

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### Smallest sphere containing three tetrahedra?

What is the smallest possible radius of a sphere which contains 3 identical plastic tetrahedra with side length 1?

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### Binary codes with upper and lower bound on pairwise distance

The Gilbert-Varshamov bound provides a lower bound for codes of length $n$ with minimum pairwise distance (say $\frac{n}8$). If we wish for the codes to also have pairwise distances bounded above (say ...

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### Upper bound of special Cheeger constant on $(S^2,g)$

$(S^2,g)$ is 2-dimensional sphere with Riemannian metric.The Cheeger constant of $(S^2,g)$ is
$$
h(S^2,g)=\inf_{\gamma} \frac{|\gamma|_g}{\min\{|A_1|_g, |A_2|_g\}}
$$
take the infimum over all closed ...

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### How does the extra rope length of this link/tangle scale with the inner triangle size?

The symmetric chiral link made of three long intertwined/linked/tangled flexible ropes of radius 1 shown in the figure, whose 6 ends all lie in a plane at spatial infinity and which are pulled ...

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### Planar sections of convex sets in Cartan-Hadamard manifolds

Let $X$ be a convex set in Euclidean space $\mathbf{R}^n$ and $p\in\mathbf{R}^n$ be a fixed point. Then any plane $\Pi$ passing through $p$ intersects $X$ in a convex set. Conversely, this property ...

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### An order statistics problem with some interesting geometry

Let $a_n$ be a given sequence of positive numbers, and $X_n$ a sequence of independent random variables with each $X_n$ uniformly distributed on $[0, a_n]$.
Question: Let $N \geq 2$ be an arbitrary ...

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### Need help understanding the geometry of a particular building structure

$\DeclareMathOperator\SL{SL}$I’m not primarily a geometer, so apologies if this question is worded poorly. I’ve been looking at asymptotic cones of connected semisimple Lie groups with at least one ...

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### A formal inquiry of geometric-problem solving

Let $\Lambda$ be a finite set. Let $\mathcal{L}$ be a finite collection of lines on a plane $X$. Then, define $X^*(\mathcal{L}) = \bigcup_{L_\alpha\neq L_\beta} L_{\alpha}\cap L_{\beta}$ to be the ...

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### Optimal packing and covering of a triangle with squares

We continue from Another variant of the Malfatti problem.
Given a triangle T and a number n, how to cover it with n squares (of possibly different dimensions) such that the sum of the areas (...

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### Arcular triangle inequality

Is it true that if inside a circular segment $S$, with vertices $a$ and $b$, we take two circular arcs, one from $a$ to $c$ and the other from $c$ to $b$, then the sum of the lengths of these two arcs ...

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### A property for maps between metric spaces

Let $X, Y$ be metric spaces with distance functions denoted by $d_X, d_Y$ respectively. Consider a map $f \colon X \rightarrow Y$. I am interested in the following property: for every $x,y,z \in X$, ...

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### What is the state of progress on this problem about continuous functions from spheres to Euclidean space?

In the 1954 paper Continuous Functions From Spheres to Euclidean Spaces, author Chung-Tao Yang cites the following problem:
Problem 1: Given a (continuous) map $f$ of an $(m+n-2)$-sphere $S^{m+n-2}$ ...

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### Snakes on a plane

A sleeping bag for a baby snake in $d$ dimensions (no, really) is a subset of $\mathbb{R}^d$ which can cover (via translation and rotation) every (piecewise-smooth for concreteness) curve of unit ...

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### Qualitative values between two electrons in an atom or how to interpret these values?

This question is a little bit trying to understand physics through geometry of simplex:
Let $E_{i,j}$ be the ionization energy in times the number of hydrogen ionization energy for an element with ...

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### Another variant of the Malfatti problem

We try to add to A Variant of the Malfatti Problem
As stated in the Wikipedia entry on Malfatti circles, it is an open problem to decide, given a number $n$ and any triangle, whether a greedy method ...

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### Algorithm for grouping tetrahedra from Voronoi diagram

I have a set of 3D Voronoi generator points and their neighbouring points, which, when connected, should result in a Delaunay tetrahedralization. However, I'm having a hard time implementing this. My ...

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### Sufficient condition for existence of a closest-point projection from a neighborhood onto a subset of a Riemannian manifold

Let $(M,g)$ be a Riemannian manifold and let $N$ be a subset of $M$.
On one hand, it is well known that if $N$ is an embedded submanifold of $M$, then it admits a tubular neighborhood, and, ...

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### Relation of geometric and polyhedral convergence

By Proposition 3.10(i) of Jorgensen and Marden's 1990 Algebraic and geometric convergence of Kleinian groups, "[A] sequence $\{G_n\}$ of Kleinian groups converges geometrically to a Kleinian ...

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### Are there four dimensional generalizations of the Reuleaux triangle and other solids of constant width? [closed]

Is there a four dimensional generalization of the Reuleaux triangle? What is it called, and what properties does it have?
Thank you!

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### Billiard circuits in pentagons

A billiard circuit in a convex $n$-gon is a closed billiard path
of $n$ segments reflecting from consecutive edges of the polygon.
Every regular $n$-gon has such a billiard circuit:
Recently a ...

5
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1
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### Expected value of the length of the shortest non-zero vector in a lattice?

$\DeclareMathOperator\SL{SL}$What is the expected value of the length of the shortest non-zero vector in a (unimodular) lattice? I.e., let $G=\SL_n(\mathbb{R})$ with Haar measure $\mu$, $\Gamma=\SL_n(...

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### Root system terminology

Let $\Phi$ be a root system. In a paper I'm writing, I need to work with subsets $\Phi' \subset \Phi$ satisfying the following two conditions:
For all $\lambda_1,\lambda_2 \in \Phi'$ and $c_1,c_2 \...

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### Where to find English translation of Pansu's paper from Ann. Math?

Where can I find English translation of the following paper?
P. Pansu,
Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. (French. English summary) [Carnot-...

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### Unusual parameterization of the ring Dupin cyclide

I discovered the following by playing with the formulas given in the paper Sculptures in $S^3$ by Schleimer and Segerman.
First, define the following parameterization of the Clifford torus:
$$
p(\...

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1
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### How to bulge out the curved edges of the stereographic tesseract?

You probably already saw such a representation of the tesseract:
I did something similar on my blog for the truncated tesseract:
The vertices in 3D are the stereographic projections of the original ...

4
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### Upper bounds on the Gromov–Hausdorff distance using persistent homology

In persistent homology theory, stability theorems are important to show that the topological signatures extracted are stable under small changes. A key result is the following bound on the bottleneck ...

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### Is there a pyramid with all four faces being right triangles? [closed]

If such a pyramid exists, could someone provide the coordinates of its vertices?

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### Algorithm to dissect a polygon into a minimum amount of rectangles, conditioned on a maximum overlap

I have the following problem, I have a problem regarding concave polygons. I want to write code to cover any polygon with a minimum amount of rectangles that are allowed to overlap and have no fixed ...