The study of geometry of manifolds without appealing to differential calculus. It includes studies of length spaces, Alexandrov spaces, and CAT(k) spaces. The techniques are often applicable to Riemannian/Finsler geometry (where differential calculus is used) and geometric group theory. For ...

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Linear bound on angles in an euclidean triangle.

I am trying to understand a proof in the book of Burago "A Course in metric geometry" (Lemma 10.8.13 page 383). I have difficulties with a certain inequality for the angle of euclidean triangles: ...
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52 views

Existence of CAT(0)-metrics

Given a metrisable, contractible topological space. Under which conditions exists a CAT(0)-metric compatible with the given topology?
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38 views

Literature on ellipses

I'm looking for good literature/sources that I can cite for working with ellipses, particularly concerning the overlap of 2 or more ellipses but also other calculations with ellipses (e.g. dividing ...
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51 views

How to show the Poincaré disk is hyperbolic for some $\delta$

I am trying to prove that the Poincaré disk, $\mathbb{D}$, is $\delta$-hyperbolic with respect to the slim triangle definition for hyperbolicity. I have been stuck for a while on where to begin, ...
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42 views

vector field distance

If we have the vector fields $X=(1,0)$ and $Y=(0, a(x))$ with $a(x)=0$ for $x\lt 0$ and $a(x) =1$ for $x \ge 0$, how can we write explicitly the distance? We have 3 cases: if we have $d(P,Q)$ where ...
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60 views

Gromov-Hausdorff distance between a “Line segment” and a “Zylinder”

I want to prove the following statement: For $X= S^1 \times [0,1]$ and $Y= [0,1]$ with the intrinsic metrics one has $d_{GH}(X,Y) = \frac{\pi}{2} $ where $d_{GH}$ denotes the Gromov-Hausdorff ...
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45 views

Lipschitz distance

The Lipschitz distance between two metric spaces is defined by $$d_{\mathcal L}(X, Y) = \inf_f \log(\{\max\{\text{dil}(f), \text{dil}(f^{-1})\})$$ where $$\text{dil}(f) = \sup_{x, y \in X} \frac{ ...
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20 views

Strainers in Alexandrov spaces

I am reading the section on Strainers in Burago, Burago and Ivanov's book "A Course in Metric Geometry". I have been struggling with the proofs of some of the lemmas. On Lemma 10. 8. 13, the authors ...
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30 views

Maximal Hausdorff dimension of the inverse image of a point

Let $f \colon [0,1]^2 \to \mathbb{R}$ be an arbitrary continuous function. I was wondering the following: Does there exist a point $a \in \mathbb{R}$ such that $f^{-1} \{a\}$ has Hausdorff dimension ...
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34 views

Squaring the plane with consecutive integer squares. And a related arrangement. [duplicate]

Q1. I was fiddling around with squaring-the-square type algebraic maths, and came up with a family of arrangements of $n^2 $squares, with sides $1,2\ldots n^2$ (n odd). Which seems like it would work ...
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59 views

Equivalent Definition of a Hermitian Metric on an Almost Complex Manifold?

For an almost-complex manifold $M$ with almost-complex structure $J$, we say that a metric $h:T_p(M;{\mathbb R}) \times T_p(M;{\mathbb R}) \to {\mathbb R}$ is Hermitian if it holds that $$ ...
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Gromov-Hausdorff convergence to a circle

I am working on the book A course in metric geometry written by D. Burago, Y. Burago and S. Ivanov, and more precisely on exercice 7.5.9: Exercice: Let $\{X_n\}$ be a sequence of compact length ...
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54 views

How much close should two spaces be in the Gromov-Hausdorff distance to be homeomorphic?

Here I am considering the Gromov-Hausdorff convergence for metric spaces. I know two compact metric spaces are isometric if and only if $d_{GH}(X,Y)=0$, where $d_{GH}$ denotes the Gromov-Hausdorff ...
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65 views

Isometry groups acting transitively

Let $X$ be a metric space and $G$ be its group of isometries. 1) Is it true that $G$ acts on $X$ transitively? If so, where can I find a proof? If not, how can one characterize those $X$ for which ...
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48 views

vertex representation and half-space representation of a polytope

i'm not a mathematician. So, my question may be stupid. Hope that it won't make you angry. Is there any mathematical relationship between the vertex representation and the half-space representation ...
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1answer
77 views

Sylvester-Gallai Theorem

How is this theorem used in applications? I've been searching for it on the web but can't seem to find. Only to "correct codes". Can someone please give a few simple examples? /lost student
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225 views

Fast calculation of the area of intersection between a sphere and a cylinder

In my current research, I am looking at calculating the local porosity of a porous media in cylindrical coordinate (notably, two co-centric cylinders). To obtain an accurate approximation, I need to ...
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1answer
58 views

Find a ruler in Taxicab plane

In the Taxicab Plane, find a ruler $f$ with $f(P)=0$ and $f(Q)>0$ for the given pair $P$ and $Q$: $1. \ P = (2,3), Q=(2,-5)$ $2. \ P= (2,3), Q = (4,0).$ The definition of ruler is ...
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66 views

characterization of uniquely geodesic normed vector spaces

I am trying to understand the proof of Proposition 1.1.6 in Bridson-Haefliger. They deal with the notion of geodesics in metric spaces, as per the definition here: ...
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85 views

Calculating the area and length of sets using a Riemannian metric on the sphere

Let $S^2\subseteq \mathbb{R}^3$ be the unit sphere. Let's define the Riemannian metric to be $d(x,y)=\angle(x,y)=\arccos(x,y)$. Calculate the area and circumference of the ball $B(x,R)=\left\{y\in ...
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55 views

A combinatorial action of a discrete group is proper if and only if it has finite vertex stabilizers

First, let me fix some definitions. The action of a group $G$ on a topological space $X$ is proper if for every compact subspace $K \subseteq X$ the set $\{g \in G \ | \ g K \cap K \neq\varnothing ...
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77 views

Is a Banach space $X$ Lipschitz equivalent to the metric quotient $X/B$, where $B$ is the closed unit ball?

Recall that the metric quotient $X/B$ is defined as follows: first we consider the equivalence relation $\sim$ on $X$ that identifies all points of $B$, then we define on the set of all equivalence ...
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38 views

The characterization of asymptotic dimension

Let X be a metric space. The following conditions are equivalent (a)asdimX = n (b)n is the smallest integer such that for every R > 0 there exists n + 1 families Ui i=0,1,2,...,n, and S > 0 such ...
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40 views

Existence of a geodesic in a complete separable metric space

If I have $X$ a complete separable metric space, $x, y \in X$ arbitrary points, how can I define a constant speed geodesic, i.e. a continuous map $g : [0,1] \rightarrow X$ such that $$ d(g(t), g(s)) = ...
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92 views

Show that the distance between these two sets is not bounded.

I have a homework question that asks: "Consider the curve $\gamma : [1, \infty] \to \mathbb{R}^2$ defined by $\gamma (t) = \langle t \cos (\ln t), t \sin (\ln t) \rangle$. Show that this curve is ...
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1answer
107 views

Is a uniquely geodesic space contractible? I

Is a uniquely geodesic space contractible ? We assume in addition that closed metric balls are compact. A post without this extra assumption is here.
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Finding an invisible circle by drawing another line

A friend of mine taught me the following question. He said he found it in a book a few years ago. Though I've tried to solve it, I'm facing difficulty. Question: You know on a plane there is an ...
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81 views

isometries of the sphere

There is a theorem by Pogorelov that if a $C^2$ surface $M$ in $\mathbb{R}^3$ is isometric to the unit 2-sphere, then $M$ is itself (a rigid motion of) the sphere. What is known about isometric ...
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29 views

Standard nomenclature for certain embedded tori

The standard embedded torus, parametrized by longitude $\alpha$ and latitude $\beta$ is given by $$ \begin{align} x & = (R+\cos\beta)\cos\alpha, \\ y & = (R+\cos\beta)\sin\alpha, \\ z & = ...
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80 views

Group acting by isometries on a length space

I am reading the book A course in metric geometry by Burago, Burago and Ivanov. I have some difficulties with an exercise 3.4.6 on page 78. The exercise is the following: Let a group $G$ act by ...
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54 views

Finding a closed curve $\gamma$ on a Torus such that $\gamma$ is not pseudo-Anosov

Let g denote genus and n denote number of punctures. According to Kra's construction in Pseudo-Anosov theory for surfaces, if S is an orientable surface such that $3g+n>3$ and $\gamma \in ...
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102 views

What does “spherical convex fuction” mean

Let $M$ be a riemannian manifold. A function $f: M \mapsto \mathbb{R} $ is called spherical convex, if \begin{equation} \sin(\lvert xz \rvert) f(y) \leq \sin(\lvert xy\rvert) f(z) + \sin( \lvert ...
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How to show that $\mathbb R^n$ with the $1$-norm is not isometric to $\mathbb R^n$ with the infinity norm for $n>2$?

Could you please give me a hint to prove that $\mathbb{R}^n$ with the 1-norm $\lvert x\rvert_1=\lvert x_1\rvert+\cdots+\lvert x_n\rvert$ is not isometric to $\mathbb{R}^n$ with the infinity-norm ...
5
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1answer
400 views

cyclic polygons & trigonometry

At one vertex of a pentagon inscribed in a circle of unit diameter (unit diameter, not unit radius) let the angles between adjacent diagonals be $\alpha,\beta,\gamma$, at the next, ...
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1answer
394 views

On nonconvex cones over compact convex sets in Hadamard spaces

Discussion http://mathoverflow.net/questions/6627/convex-hull-in-cat0 indicates the convex hull of a finite set can fail to be closed in a complete Hadamard space. Hence the following question should ...
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1answer
161 views

Unit speed geodesics

This might be a very trivial question so bare with me. If $(X,d)$ is a length space we define a unit speed geodesic to be a path $\gamma:[0,1]\to X$ for which \begin{align*} ...