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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25 views

Given a non-ideal hyperbolic triangle and the Euclidean comparison triangle with equal side lengths, are the interiors of the two bi-Lipschitz?

Fix three finite real numbers $p,q,r > 0$. Up to isometry, there is a unique 2-simplex $\Delta$ in the Euclidean plane bounded by a geodesic triangle with these three reals $p,q,r$ as side-lengths. ...
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45 views

On the centroid of a triangle

There's three different ways to see a triangle in the Euclidean plane: as three non-collinear points, say $A$, $B$, $C$; as the line segments connecting the three points, that we can parametrize as a ...
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1answer
18 views

CAT(K) Finsler manifolds.

I was wondering if the following is true (and common knowledge): Let $(M,F)$ be a Finsler manifold. Let d be the induced distance by the norm in the usual sense. That is, $d(x,y)=\inf${lenghts of ...
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36 views

A measure on the space of probability measures

I've been reading about optimal transport and it's connections to geometry. At some point one has to study a bit of the structure of the space of probability measures, $\mathcal{P}(X)$, (over a metric ...
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1answer
21 views

Invariance of ball size under translation on the pseudosphere

On the pseudosphere, do the volume of metric balls and area of metric spheres change with the centerpoint? (I'm asking because they don't on the sphere.)
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22 views

Adjacent angle Theorem in planes of constant curvature - easy geometric proof?

I am trying to proof the Adjacent angle Theorem in planes of constant curvature (2-Sphere, euclidean plane, hyperbolic plane) i.e given 4 points $a,b,c,d$ such that $d$ is lying on a shortest curve ...
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88 views

How to build the smallest regular n-sided polygon that covers an (n-1)-sided polygon?

I want to build a figure that contains seven regular polygons, from a triangle up to a nonagon, where each n-sided polygon covers, with the minimal area possible, the n-1 sided one. An added ...
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1answer
38 views

why are CAT(0) spaces contractible?

In the book of Bridson and Haefliger it is said that 'it follows easily' from what they proved before. Does anyone know of a rigorous proof that CAT(0) spaces are contractible?
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91 views

general formula for volume of a simplex?

I am looking for a general formula to calculate the volume of a euclidean simplex in any number of dimensions. On Wikipedia I found that a formula similar to Heron's formula can be applied to ...
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27 views

Ultraproduct of a metric space

I am currently trying to understand "Curvature bounded below: a definition a la Berg--Nikolaev" by Nina Lebedeva and Anton Petrunin. They start with a complete, intrinsic metric and space $X$ and say ...
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48 views

Closure of a region and shortest path

Let $\Omega$ be a region in $\mathbb{R}^2$ and $\overline{\Omega}$ be closure of $\Omega$. Is it true that between every two points $x,y \in \overline{\Omega}$ exists shortest path (lenth of path ...
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44 views

How to calculate the center of a regular polygon?

What is the formula for the center of an n-edge regular polygon that has the given segment as its edge? So, given a segment AB, ...
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1answer
49 views

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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1answer
60 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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40 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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2answers
71 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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0answers
73 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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1answer
78 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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36 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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1answer
34 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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35 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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1answer
74 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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330 views

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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62 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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111 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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66 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
84 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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1answer
242 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
64 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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1answer
83 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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1answer
122 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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63 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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88 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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44 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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1answer
46 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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95 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
121 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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1answer
92 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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34 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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1answer
94 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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0answers
60 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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1answer
108 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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684 views

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
430 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
410 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
197 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*} ...