# Questions tagged [mg.metric-geometry]

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

2,294 questions

**5**

votes

**1**answer

74 views

### Sum of squared nearest-neighbor distances between points in a square

Let $\square_2=\{(x,y): 0\leq x, y\leq1\}$ be the unit square in $\mathbb{R}^2$. Take $n>1$ points $P_1, \dots, P_n\in\square_2$.
Denote the distances $d_j=\min\{\Vert P_k-P_j\Vert: k\neq j\}$, ...

**21**

votes

**15**answers

3k views

### Geodesics on the sphere

In a few days I will be giving a talk to (smart) high-school students on a topic which includes a brief overview on the notions of curvature and of gedesic lines. As an example, I will discuss flight ...

**2**

votes

**0**answers

74 views

### The typical amount of lattice points in a set as dimension goes off to infinity

I have encountered a geometry-of-numbers-type problem when calculating an entropy in a lattice communications scheme:
$L^{(n)}$ is a lattice uniform over all those in $\mathbb{R}^n$ with base ...

**3**

votes

**0**answers

107 views

### Can bellows make loops?

Can flexible polyhedron (hyperbolic or euclidean) have non-simply connected configuration space not containing singular polyhedra?

**4**

votes

**1**answer

89 views

### non-proper parabolic isometries of hyperbolic spaces

In his seminal paper on hyperbolic groups (see Section 8.1) Gromov defines an isometry $f$ of a hyperbolic space $X$ to be parabolic if the orbit of any point $x\in X$ under the action of $\langle f\...

**8**

votes

**1**answer

215 views

+50

### Packing regular tetrahedra of edge length 1 with a vertex at the origin in a unit sphere

Consider the following problem:
How many regular tetrahedra of edge length 1 can be packed inside a unit sphere with each one has a vertex located at the origin?
The answer is at least 20, forming ...

**5**

votes

**1**answer

114 views

### Quantitative upper bound on mean curvature of an isometric embedding

By Nash embedding theorem, any complete Riemannian manifold $M$ can be isometrically embedded in $\mathbb{R}^N$, for sufficiently large $N$.
The proof of the theorem is quite involved, and it is not ...

**3**

votes

**2**answers

151 views

### How can the visible surface area of a box at any angle be found? [on hold]

If a box with dimensions x, y, and z exists, an xyz axis is set with a camera having no rotation relative to it, and this box is rotated angle_x degrees, angle_y degrees, and angle_z degrees relative ...

**4**

votes

**0**answers

82 views

### How to think about dual space of a certain space of Lipschitz functions

Consider the following Banach space (for concreteness):
$$X=Lip(\bar{\mathbb{B}}^n)=\{f\in C^0(\bar{\mathbb{B}}^n): \Vert f \Vert_L<\infty \}$$
where
$$
\bar{\mathbb{B}}^n=\{\mathbf{x}\in \mathbb{...

**5**

votes

**1**answer

246 views

### Functions that map open balls to open balls of different radius?

For $n \geq 2$ we say a continuous function $f: \mathbb R^n \to \mathbb R^n$ such that the image of any bounded open ball is a bounded open ball of different radius is a balloon function.
...

**5**

votes

**2**answers

139 views

### Origin of term Ahlfors-David regular

Much of the literature on analysis in metric spaces makes use of an assumption called Ahlfors regularity or Ahlfors-David regularity. Let $q>0$. A metric space $(X,d)$ is Ahlfors(-David) $q$-...

**4**

votes

**1**answer

297 views

### Spreading $n$ points in $\{0,1\}^n$ as far as possible

Given a positive integer $n$, the Hamming distance $d^H_n(x,y)$ of $x,y\in \{0,1\}^n$ is defined by $$d^H_n(x,y) = |\{k\in\{0,\ldots,n-1\}: x(k)\neq y(k)\}|.$$
We say that a positive integer $s$ is $...

**4**

votes

**1**answer

129 views

### Flat norm metrizes the weak* topology

I've come across the following statement in literature (without proof or reference) about the flat norm of currents
$$
F(T) = \sup \{ T(\omega) : \omega \in D^k(U), |\omega(x)| \leq 1, |d\omega(x)| \...

**25**

votes

**3**answers

578 views

### What is the structure preserved by strong equivalence of metrics?

Let $X$ be a set. Then we can define at least three equivalence relations on the set of metrics on $X$. We say that two metrics $d_1$ and $d_2$ are topologically equivalent if the identity maps $i:(...

**6**

votes

**0**answers

85 views

### Adjoint of the Hodge de Rham star operator under the integral pairing

Given a Riemannian manifold $(M, g)$ of dimension $n$, the Hodge star operator $\star: \Omega^k(M) \to \Omega^{n-k}(M)$ is defined. What is the (formal) adjoint of $\star$ under the integration ...

**15**

votes

**2**answers

1k views

### Integrating over a hypercube, not a hypersphere

Denote $\square_m=\{\pmb{x}=(x_1,\dots,x_m)\in\mathbb{R}^m: 0\leq x_i\leq1,\,\,\forall i\}$ be an $m$-dimensional cube.
It is all too familiar that $\int_{\square_1}\frac{dx}{1+x^2}=\frac{\pi}4$.
...

**-1**

votes

**0**answers

34 views

### Order Statistics and High Dimension Geometry

Suppose that I have iid random variables $\mathrm U_n \sim \mathrm U(0,1)$. Then, for $\mathrm Y_m$ defined as,
$$\mathrm Y_m = \min_{n \in [1,m]} \mathrm U_n$$
it is easy to compute $\mathbb{E}[\...

**1**

vote

**0**answers

88 views

### Some Problems On Apollonian Gasket

Since 2013, I found Some problems on Apollonian Gasket as following. These problem also is higher level of Eppstein Point. I am looking for a proof of one of these problems:
Let three $(A)$, $(B)$, $(...

**1**

vote

**1**answer

101 views

### 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 $L$-...

**2**

votes

**0**answers

126 views

### A problem of four conics

I found a remarkable theorem of four conics as follows some years ago. But it has no proof; I am looking for a proof:
Theorem: Take three conics. Suppose that each of them touch a fourth conic at two ...

**11**

votes

**3**answers

255 views

### Mean maximum distance for N random points on a unit square

Following up on Mean minimum distance for N random points on a one-dimensional line and Mean minimum distance for N random points on a unit square (plane), I have the following questions.
Given N ...

**4**

votes

**1**answer

85 views

### Conformally flat homogeneous spaces

Let's say we have a homogeneous space $H\backslash G$.
Is it possible to tell whether this homogeneous space admits a conformally flat metric just from its group structure?
I am particularly ...

**5**

votes

**1**answer

108 views

### Is a symmetric, parallel (0,2)-tensor a metric?

I'm interested in affinely connected spaces, on which a metric is not necessarily defined, i.e. $(\mathcal{M},\Gamma)$. Since (as a physicist) my goal is to consider a generalized model of gravity, I ...

**1**

vote

**1**answer

72 views

### How do I find the correct distance spacing for distributing equidistant points on a sphere of a given diameter?

How do I find the correct distance spacing for distributing equidistant points on a sphere of a given diameter?
I don't need to fill the sphere with equidistant points. I just need less than a ...

**4**

votes

**0**answers

78 views

### Optimal placement of points on sphere to minimize size of projections

While studying some physical problem we stumbled across a geometrical problem, to which we could not find any solution.
Consider a unit sphere in 3D and a collection of $N$ points on its surface.
Now ...

**3**

votes

**0**answers

73 views

### Reference of generalized isometries

I'm wondering if these objects have a name or are studied. No one around me knows, so I thought to ask here.
Let $\Phi:\mathbb{R}^d\rightarrow \mathbb{R}^d$ be $C^2$-diffeomoprhism, and fix $p \geq ...

**5**

votes

**0**answers

118 views

### Does $G$ act 2-transitively on its Bruhat-Tits building?

Let $k$ be a finite extension of $\mathbb{Q}_p$ and let $G$ be a semisimple Lie group over $k$. We consider the action of $G$ on its Bruhat-Tits building $X$.
Question: If $x,y,x',y'$ are vertices, ...

**1**

vote

**1**answer

78 views

### A question on a special “metric”

Suppose we have a function $F: [a,b]^n \to \mathcal{M}_{n \times n }(\mathbb{R})$ where $\mathcal{M}_{n \times n }(\mathbb{R})$ is the space of $n \times n$ real matrices, a compact set $B \subset \...

**6**

votes

**2**answers

264 views

### What is the geometric meaning of one Riemannian metric bigger than the other one on a smooth manifold?

Gromov conjectured in 1985 and LLarull proved in 1998 that: If $g > g_0$ on the sphere, then there exists some point p on the sphere with $Sc(p) < Sc_0(p)$. Here $g, g_0$ are Riemannian metrics ...

**2**

votes

**0**answers

32 views

### Rough classification of segment metrics

For a dynamical project, I need to bound uniformly a quantity defined from a metric on the segment $I=[0,1]$ over all metrics inducing the usual topology. The details do not matter too much, I wonder ...

**0**

votes

**0**answers

44 views

### Does lattice mod preserve direction?

For high enough dimension $n$ there are lattices $L_n$ in $\mathbb{R}^n$ whose Voronoi partition's base regions encompass all but a negligible proportion of a $(1-\varepsilon)$-ball, and also nearly ...

**-1**

votes

**1**answer

40 views

### Characterisation of a superset of the simplex

Does there exist a nice description of the following set:
\begin{equation}
A:=\left\lbrace x\in\mathbb{R}^{n}\ \colon\ 0< x_{i}-\bar{x}+\frac{1}{n}< 1\ \text{for} \ i=1,\dots,n\right\rbrace,
\...

**7**

votes

**3**answers

329 views

### Sectional curvature of leaves of foliation

Given a $k$- dimensional foliation $F$ of a riemannian $n$-manifold $M$, with the property that the leaves of the foliation have constant sectional curvature $s$, for some $s$, is it true that $M$ ...

**21**

votes

**6**answers

2k views

### Smooth functions on sphere

Let $u$ be a smooth function defined on the unit sphere $S^2$. Assume $u$ has two local maxima, two local minima, and two saddle points (a total of 6 critical points). Does there exist a plane $P$ ...

**2**

votes

**2**answers

79 views

### Broken geodesic in Finsler polyhedral space

Here we assume that all norms has only one geodesic, i.e. locally
minimizing, between any two points.
Example : In $\mathbb{R}^2$, a line $y=kx,\ k>0$ divides
$\mathbb{R}^2$ into two regions. We ...

**6**

votes

**0**answers

105 views

### Can scalar curvature and diameter control volume? Round 2

This is a follow up to a question by Yiyue Zhang. Can scalar curvature and diameter control volume?
The original question asked whether scalar curvature bounds and small diameter bounds were enough ...

**14**

votes

**3**answers

328 views

### What is known about sufficient conditions for the rigidity of a convex surface?

A convex surface is a connected open subset of the boundary of a convex body in $\mathbb{R}^3$.
An "infinitesimal bending" of a convex surface $S$ is a deformation of $S$ given by a velocity field $v:...

**5**

votes

**2**answers

110 views

### Function as sum of distances over a connected, compact metric space

If $X$ is a connected, compact metric space with distance function $d : X^2 \rightarrow \mathbb{R}^+$, is it true that there exists a positive real number $a$, dependent on $X$ and $d$, such that for ...

**18**

votes

**1**answer

307 views

### Gluing hexagons to get a locally CAT(0) space

I believe that there are four ways to glue (all) the edges of a regular Euclidean hexagon to get a locally CAT(0) space:
The first two give the torus and the Klein bottle, respectively. What are the ...

**2**

votes

**0**answers

139 views

### An new equilateral triangle related to the Morley triangle

Morley equilateral triangle is the nice theorem in Eulidean Geometry. I found an equilateral triangle and a group circle related to the Morley triangle and angle trisectors:
Let $ABC$ be a triangle ...

**1**

vote

**0**answers

63 views

### Some hypersurface has a positive second fundamental form potentially

Notation : $r^2=x^2+y^2$.
Exercise : Define $$F_\sigma (x,y)= (f_\sigma
(x,y),\frac{x^2-y^2}{2},xy)$$ where $$f_\sigma (x,y)=(1- \sigma^2
r^2)(x-\sigma x^3,y-\sigma y^3)$$
Define $ G_\sigma: \mathbb{...

**6**

votes

**0**answers

95 views

### Do manifolds with non-negative Ricci curvature allow bi-Lipschitz embeddings into Euclidean spaces?

QUESTION: Let $n$ be a natural number. Is it true that there exist $N(n), D(n) > 0$ such that any complete $n$-dimensional Riemannian manifold of nonnegative Ricci curvature can be embedded into $N$...

**4**

votes

**2**answers

148 views

### largest diameter of intersection of two balls

Two closed balls with a common radius are positioned so that the centre of either ball is on the boundary of the other.
I am interested in the extremal diameter of their intersection, in an arbitrary ...

**4**

votes

**0**answers

130 views

### Non-algebraic quasi-isometric embeddings

What are examples of finitely generated groups $\Gamma$ and $\Lambda$ such that the metric space $\Lambda$ embeds into $\Gamma$ quasi-isometrically but such that $\Lambda$ is very much not a subgroup ...

**3**

votes

**0**answers

206 views

### Are these points known? [closed]

Let $ABC$ be a triangle and $P$ be a point on the plane, $PA$, $PB$, $PC$ meet $BC$, $CA$, $AB$ at $A'$, $B'$, $C'$ respectively.
From my construction by GeoGebra, I found two special points as ...

**1**

vote

**0**answers

77 views

### DIstance on a Riemannian manifold [closed]

Given a Riemannian manifold $(M,g)$ is it possible to calculate the distance between two points on this manifold. Is it possible the inverse? That means: given a formula of the distance, for example:
...

**6**

votes

**0**answers

90 views

### Geometric mean of three or more positive definite matrices

The geometric mean of two positive definite (Hermitian) matrices of same size is defined by $$A\natural B := A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},$$equivalently,
$$A\natural B =(BA^{-1})^{1/2}A=A(A^...

**4**

votes

**0**answers

149 views

### Uniqueness of the boundary of a hierarchically hyperbolic group

Hierarchically hyperbolic groups and spaces (HHG and HHS for short) were defined by Behrstock, Hagen and Sisto (see here and here). Examples include mapping class groups, Right angled Artin groups, ...

**13**

votes

**0**answers

223 views

### Why are the medians of a triangle concurrent? In absolute geometry

This fact holds true in absolute geometry, and I would like to see an elementary synthetic proof not using the classification of absolute planes (Euclidean and hyperbolic planes) and specific models. ...

**3**

votes

**1**answer

133 views

### Are $CAT(0)$-polygonal complexes median spaces?

A median space is a metric space $X$ for which for any three points $x, y , z \in X $ there exists a unique point $m$ such that $d(x,m)+ d(m, y)= d(x , y ), d(x,m)+ d(m, z)= d(x , z ), d(y,m)+ d(m, z)=...