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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58 views

Alternative Geometries

In our world, the distance between two points (in 2d) is defined as $\sqrt{(\Delta x)^2 + (\Delta y)^2}$. Suppose that in an alternative geometry, it was defined as $\sqrt[p]{|\Delta x|^p + |\Delta ...
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2answers
24 views

Hausdorff dimension of homeomorphic compact metric spaces

Are there examples of homeomorphic compact metric spaces of different Hausdorff dimension? If yes, are there sufficient conditions on the spaces which would imply the equality of Hausdorff ...
2
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1answer
43 views

Centralizers in mapping class groups

According to Nielsen-Thurston classification, given a closed surface $S$, the elements of the mapping class group $\mathrm{MCG}(S)$ lie in three categories: periodic, reducible and pseudo-Anosov. ...
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16 views

A complete metric space with approximate midpoints is intrinsic

Let $(X,d)$ be a metric space. For $x,y\in X$, define $A_{xy}$ the set of curves (the domain is supposed to be $[0,1]$) joining $x$ with $y$. For $\sigma\in A_{xy}$, define its length as ...
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44 views

How can the notion of “two curves just touching” (vs. “two curves intersecting”) be expressed for a given metric space?

A popular introductory description of a "tangent (in geometry)" is presented as "the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the ...
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56 views

Given a spacetime in terms of values of Lorentzian distance, how to determine whether it is flat?

Yesterday I learned that geometric relations between events can be characterized generally (and up to a common non-zero factor) in terms of their pairwise "Lorentzian distance, $d_{\ell}$", which ...
3
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75 views

Taylor expansion of the square of the distance function

Give a smooth Riemannian manifold $(M,g)$, (i) how can one compute Taylor expansion of the square of the Riemannian distance function $d^2(x,x_0)$ at $(x',x_0)$? I've tried to use $dist=\int ...
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30 views

Connections between Level Sets and convex/concave Functions

I am interested in the connection between level sets and convex/concave functions on Manifolds. For example if a Function is convex (concave) all sublevel (superlevel) sets are convex. Is the ...
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42 views

Concavity of distance function in $\mathbb{R}^n$ or determinant of $(x^T \cdot x)$

I would like to compute the concavity of the distance function in $\mathbb{R}^n$. Let $ f(x) =- \Vert x \Vert $ in $\mathbb{R}^n$. Then $\nabla_xf=- \frac{x}{\Vert x \Vert}$. And ...
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1answer
42 views

Are Homogenous countable complete metric spaces always discrete?

Let $M$ be a countable complete metric space such that the group of isometries of $M$, $Iso(M)$ acts transitively on the points in $M$. Does it follow that the topology induced by the metric is the ...
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63 views

How to find the “area” of a box (hyperrectangle) projected onto a ball (hypersphere)?

What is the "area" of the general $n-1$ - box (hyperrectangle) projected onto the surface of the general $n$ - ball (hypersphere)? I'm just curious.
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95 views

Moving circular disk between two parallel sinusoidal curves

Find the largest radius of the circle that can be "rolled" between the curves $y = sin(x)$ and $y = sin(x)+1$. After two weeks of research, I finally give up.
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37 views

Prerequsites for working through the 2nd half of Gradient flows in metric spaces and in the spaces of probability measures

I apologize in advance if this question is too general, that is, not a request for a specific reference, but more of a request for a road map, perhaps from someone that knows the material and, in ...
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1answer
114 views

upper bound and a lower bound on the number of points that are uniformly distributed on a surface

Can I calculate an upper bound and a lower bound (or max or min) on the number of points that are uniformly distributed on a surface, knowing the area of the surface ? More precisely, I have a sector ...
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1answer
35 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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62 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
29 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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58 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
34 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.)
4
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1answer
117 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
80 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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1answer
184 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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46 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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1answer
57 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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172 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
54 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
74 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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2answers
109 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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103 views
+100

Gromov-Hausdorff distance between a line segment and a cylinder

I want to prove the following statement, where $d_{GH}$ denotes the Gromov-Hausdorff distance: For $X= S^1 \times [0,1]$ and $Y= [0,1]$ with the intrinsic metrics one has $d_{GH}(X,Y) = ...
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1answer
147 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_1, x_2 \in X} ...
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1answer
45 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
44 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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40 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
84 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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1answer
340 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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1answer
84 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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0answers
192 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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0answers
92 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
93 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
350 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
79 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
95 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
171 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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71 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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1answer
100 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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46 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
53 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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0answers
97 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 ...
2
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1answer
135 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 ...