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Questions tagged [mg.metric-geometry]

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

5
votes
1answer
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
15answers
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
0answers
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
0answers
107 views

Can bellows make loops?

Can flexible polyhedron (hyperbolic or euclidean) have non-simply connected configuration space not containing singular polyhedra?
4
votes
1answer
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
1answer
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
1answer
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
2answers
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
0answers
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
1answer
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
2answers
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
1answer
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
1answer
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
3answers
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
0answers
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
2answers
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
0answers
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
0answers
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
1answer
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
0answers
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
3answers
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
2answers
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
0answers
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
0answers
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
1answer
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
3answers
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
6answers
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
2answers
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
0answers
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
3answers
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
2answers
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
1answer
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
0answers
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
0answers
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
0answers
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
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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)=...