Questions tagged [metric-geometry]

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 questions about plain-old metric spaces, please use (metric-spaces) instead.

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27 views

Must a Geodesic Metric Space be a Length Space?

A metric space $(X,d)$ is said to be a geodesic metric space if and only if each pair of points $x,y \in X$ is connected by a geodesic, where the geodesic need not be unique. I assume that having this ...
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85 views

geodesic in metric space and in manifolds

In the book by ''Metric spaces of non-positive curvature'' by Bridson and Haefliger we have the following definition for a geodesic in a metric space: Let $(X,d)$ be a metric space. A map $c:[0,l]\...
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53 views

How can I solve the problem of convex hull

Let $\DeclareMathOperator{\Conv}{\mathrm{Conv}} C=\Conv(v_1,v_2,...,v_m) $, where $ v_1,v_2,...,v_m $ are $ m $ points in $ \mathbb{R}^n $ and 'Conv' means the convex hull. Please prove $$ \partial C=...
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Geodesics in a metric space are locally distance minimizers?

I am interested in the concept of geodesic in metric space. In wikipedia I read In metric geometry, a geodesic is a curve which is everywhere locally a distance minimizer. More precisely, a curve $γ :...
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1answer
61 views

Examples of CAT(0)-spaces that are no manifolds

I´m trying to think of some examples of CAT(0)-spaces that are not manifolds. I haven´t found any examples in the book of Bridson and Haefliger, so I have tried to come up with some own examples. I ...
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30 views

Is every submetry 1-Lipschitz?

A submetry is usually defined as a function $f:X\to Y$ between metric spaces such that, if $B(x,r)$ is the closed ball of radius $r$, we have the for every $x$ in $X$, $$ f\big(B(x,r)\big) = B \big( f(...
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133 views

Doubling of spherical triangle in $\mathbb R^3$

Here is my though process: The standard unit sphere is an Alexandrov space with curvature bounded from below by $1$. An octant of that sphere (including boundary) is a convex subset of that sphere ...
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40 views

Proper Discontinuity of $SO(2)$ action on $S^k$

The complex projective space $\mathbb{C}P^n$ can be represented as $\mathbb{C}P^n\cong S^{2n+1}/S^1$ where the elements of $(z_1,\dots,z_{2n+2})\in S^{2n+1}\subseteq \mathbb{C}^{2n+2}$ is quotiented ...
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54 views

CAT(0) inequality and CN inequality

I am currently studying CAT(0)-spaces following the book of Bridson and Haefliger: metric spaces of non positive curvature. Chapter II, exercise 1.9 is the following: Let $(X,d)$ be a geodesic metric ...
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39 views

Can All 'Dihedral Spherical Polygons' Be Simplified?

For the purposes of this question, a spherical polygon in $\mathbb S^2$ with isometric side pairings is called dihedral if: It is star-shaped with respect to some interior point; It is homeomorphic ...
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Composing projections in $\operatorname{CAT}(0)$ spaces

Let $\alpha\subset\Pi\subset M$ be geodesic $\alpha$ contained in a geodesic plane $\Pi$ in a $\operatorname{CAT}(0)$ space $M$, and for any convex geodesic subspace $X$ let $p_X:M\to X$ be the ...
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Rigidity for convex surfaces in elliptic/hyperbolic space

From Alexandrov's work we know that any metric on the sphere with lower curvature bound $\kappa$ (in the sense of Alexandrov) can be realized as a closed convex surface (i.e. boundary of a convex ...
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What is the projection of a semi-metric?

Let $X$ be a set. A semi-metric $d$ is a function $d:X\times X\rightarrow \mathbb{R}^+\cup \{\infty\}$ such that a) $d(x,y)>0 \text{ if } x\neq y$, b) $d(x,y)=d(y,x)$, c) $d(x,y)\leq d(y,z)+d(x,z)$,...
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23 views

Composition of $\varepsilon$-isometries

Let $X$ and $Y$ be metric spaces, and let $f:X\to Y$ be a function (not necessarily continuous). The function $f$ is called an $\varepsilon$-isometry, for $\varepsilon\ge 0$, if both the following ...
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73 views

Generalization of consequence of law of cosine

There is the immediate consequence of the law of cosine stating that when fixing two sidelengths of a triangle and increasing the third, the vertex angle opposite of the third side increases as well. ...
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44 views

Definition of pointed Gromov-Hausdorff convergence for metric spaces

Whereas the definition of Gromov-Hausdorff convergence for compact metric spaces seems to be standard, difference sources seem to give slightly different definitions of pointed Gromov-Hausdorff ...
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79 views

Area of a $r$-tubular neighborhood of a union of convex sets

Consider $P_i$ which is a regular $i$-gon in $\mathbb{R}^2$ and whose diameter is $1$. Define a compact set $X$ to be a union of convex hulls of copies of $P_i,\ i\geq 3$, which is in some rectangle $[...
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73 views

Almost-isometric embedding is almost surjective

I'm looking for a statement, if it exists, of the following sort: Let $X$ be a compact metric space. Let $f:X\to X$ be a function, and suppose that for some (fixed) $\epsilon>0$, for all $x,y\in X$...
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22 views

Using the Rodrigues' rotation formula to rotate a parametric surface?

I am currently trying to solve a problem that involves rotating a parametric surface in 3D space. The surface is a Cone, with parameterisation $C=r \cos(a)i + r \sin(a)j + rk$. To do this I decided to ...
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1answer
46 views

Convergence of Subsequences in Gromov's Theorem

A key theorem in metric geometry is Gromov's Compactness Theorem, which describes when a sequence of metric spaces has a Gromov-Hausdorff converging subsequence. I'm not sure I follow how this works: ...
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109 views

What is a plane figure covering all sets of diameter $1$?

Problem : (1) Show that there is a plane figure $F$ of least area which is capable of covering any plane figure of unit diameter. (2) Try to guess what is $F$. Proof of (1) : Define $\mathcal{H}$ to ...
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1answer
37 views

Find a convex polygon $P$ s.t. $P\subset Q \subset (1+\epsilon) P $.

Assume that $Q$ is a compact convex plane figure containing $R$-ball of the center origin $o$. Then prove that for any $\epsilon >0$, there is a convex polygon $P\subset Q$ s.t. $$ P\subset Q\...
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64 views

Sufficient conditions for curves with known length to lie on some Riemannian Manifold

I'm new to Riemannian geometry, trying to figure out if it's the tool I need before diving in head first. Sorry if this is a basic question. Say I have a smooth manifold $M$ of dimension $n$ with a ...
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1answer
113 views

Upper bound for the total curvature of a shortest path in the boundary of a convex polyhedron in $\mathbb{R}^3$.

Consider finitely many points in $\mathbb{R}^3$. The boundary of the convex hull is $\Sigma$. When $f_i$ is a face and $u_i$ is unit outnormal to $f_i$, then assume that $$(-u_1)\cdot u_i >\eta>...
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2answers
128 views

An involute of a shortest path in a strictly convex surface in $\mathbb{R}^3$.

Consider a smooth closed surface $\Sigma$ in $\mathbb{R}^3$ of positive Gaussian curvature, homeomorphic to a sphere. Here $U$ is the interior of the convex hull of $\Sigma$. When $c:[0,l]\rightarrow \...
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2answers
90 views

Is a pyramid uniquely determined by its edge-lengths?

I am looking for a nice/short proof of the following: The shape of a pyramid with a convex polygonal base is already uniquely determined by knowing the length of all its edges. By "knowing the ...
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1answer
34 views

Has anybody a reference for the volume of the N-1 dimensional simplex between the points N on every axis

I am looking for a reference for the volume of the simplex in N dimensions between the points N on every axis. E.g in 2 dimensions the line between the points (0,2) und (2,0). the length is $$\sqrt{2} ...
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37 views

Busemann function-like function in Riemannian manifold $\mathbb{R}^n$ is $1$-Lipschitz

With user10354138' comment, let me explain the setting (in the paper) : Consider a Busemann function in Riemannian manifold $( \mathbb{R}^n,d)$ with $\mathbb{Z}^n$-isometric action $\ast$ where $d$ is ...
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37 views

$G$-invariant metrics on homogeneous spaces $G/H$

Let $G$ be a compact metrizable group and let $H$ be a closed subgroup. Let $d$ be a compatible metric on $G$ that is left-invariant and it is also right invariant with respect to the elements of $H$. ...
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39 views

Pointwise convergence is uniform convergence under a condition

[1] pointwise convergence is uniform convergence under some condition [2] A course in metric geometry - Burago, Burago and Ivanov Question : I want to understand intuitively or visually the proof ...
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1answer
47 views

pointwise convergence is uniform convergence under some condition

When $f : \mathbb{R}^2\rightarrow \mathbb{R}$ is a nonnegative continuous function, then define $F (v) = \lim_{t\rightarrow \infty}\ \frac{f(tv)}{t}$. When $F$ is a norm on the vector space $\mathbb{R}...
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1answer
59 views

Does convexity implies contractibility in length space?

It is known that convexity implies contractibility in Euclidean spaces. I want to know whether it holds in a general length space. To be specific: Let $(X,d)$ be a length space and $A\subseteq X$. $A$...
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76 views

When is the quotient of a geodesic space again a geodesic space?

I am interested in the behavior of the quotient semi-metric on geodesic spaces, i.e. length spaces where there is always a minimal curve between two points. I used the following definition of the ...
3
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1answer
185 views

what are the inner angles of a tetragonal trapezohedron

I am making a 3D wooden lantern in the shape of a Tetragonal Trapezohedron. Think of a hollow Tetragonal Trapezohedron 3D object, where each face is a plank of wood. I am trying to determine the ...
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212 views

Example of a quasi-open set and how the capacity of the set fits in it?

While going through some research papers, I came across a result of D. Bucur where the existence of a minimizer for the general $k^{th}$ eigenvalue of the Dirichlet Laplacian among a class of quasi-...
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1answer
59 views

Length of the diagonal of a “dented triangle”

Consider a "dented triangle" like in this picture: (the angle $\alpha$ is larger than $\pi$). Compare it with an actual triangle $\triangle\bar{A}\bar{B}\bar{C}$ with sidelengths $a, b$ and $c = c' + ...
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1answer
216 views

Understanding a Proof: The square root of any metric is ptolemaic..

In Hyperbolicity, CAT(-1)-spaces and the Ptolemy Inequality, there is a short proof of a simple statement: Let $(X, d)$ be an arbitrary metric space. Then $(X, \sqrt{d})$ satisfies the Ptolemy ...
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1answer
65 views

Contractable Metric Spaces Homeomorphic to Euclidean Space

Is there a characterization of all metric spaces which are homeomorphic to a contractable subset of Euclidean space? This question is cross-referenced here.
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Existence of $1$-Lipschitz map between triangles

Consider two (Euclidean) triangles $T$ and $T'$. Let's say that $T$ majorizes $T'$ if there exists a 1-Lipschitz map that sends vertices to vertices and sides to sides (for some labeling of the ...
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1answer
65 views

Limit of points in ellipse

Consider $$c(t)=(a\cos\ (t),b\sin\ (t)),\ 0\leq t,\ a=1 > b>0$$ in $\mathbb{R}^2$ Fix $0<t_1<\pi/2$ so that $x_1=c(t_1)$ When $x_i=c(t_i)$ and $x_i$ is a positive multiple of $c'(t_{i-...
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1answer
112 views

How is $\overline{c_{\epsilon}}:=\overline{\bigcup_{n\geq 1}\left(\bigcap_{k=n}^{\infty}B(x_k,r+\epsilon)\cap K \right)}$ convex in a CAT$(0)$ space?

Let $(X,d)$ be a CAT$(0)$ space, $\{x_n\}\subset X$ be bounded and $K\subset X$ be closed and convex. Define $\varphi:\,X\longrightarrow \mathbb{R},$ by $\varphi(x)=\limsup\limits_{n\to\infty}d(x,x_n)$...
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266 views

Length is invariant under parameterization

I'm reading the book A Course in Metric Geometry by Dmitri Burago, Yuri Burago Sergei Ivanov. In page 45 it says "it is easy to see that all parametrization of a curve have equal length", but I do not ...
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96 views

Exact sequence in cohomology while studying orientability of Alexandrov spaces

I'm reading the article Orientability and fundamental classes of Alexandrov spaces with applications by Ayato Mitsuishi. There are the following definitions: for $n\geq 1$, an $n$-dimensional MCS ...
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1answer
229 views

Why the length is lower semi-continuous?

I'm reading the book A Course in Metric Geometry by Dmitri Burago, Yuri Burago Sergei Ivanov and I don't understand the proof of Proposition 2.5.17 (Page 48). More precisely, I don't know why the ...
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37 views

Every finite metric space $X$ can be embedded into $\ell_{C_\epsilon\log(|X|+1)}$ with low distortion

I have been tasked with proving that for all $\epsilon >0$ there exists some $C_\epsilon$ such that every finite metric space $X$ embeds into $\ell_{C_\epsilon\log(|X|+1)}$ with distortion at most $...
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2answers
180 views

Reference request: Introduction to Finsler manifolds from the metric geometry point of view (possibly from the Busemann's approach)

This question is a cross post from MathOverflow. I have requested the migration of the question, but unfortunately it is not possible after two months of posting. I was reading about geometry in ...
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1answer
59 views

Free faces in $M_\kappa$ polyhedral complexes.

I am reading Bridson and Haefliger's ``Metric Spaces of Non-Positive Curvature" and I am struggling with Definition 5.9 of a free face. Let $K$ be an $M_\kappa$-polyhedral complex. A closed $n$-...
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1answer
54 views

Metric space analog for manifold with boundary

I tried to define an analog of manifold with\without boundary for metric space and I wondered if this definition or a similar one exists in the literature. Let $\left(X,d\right)$ be a connected ...
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22 views

Finite number of jumps in metric for space with finite number of components

Let $(X,d)$ be a metric space with at most countably-infite many discontinuities and fix $x \in X$. For every $r \geq 0$, define the sets $A(r)$ by $$ A(r)\triangleq \{y \in X : d(x,y)=r\}, $$ and ...
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1answer
177 views

Every riemannian length structure on $\mathbb{R}^n$ is induced by a continuous function $f:\mathbb{R}^n\to \mathbb{E}^n$, to the euclidean space.

This question is a cross post from MathOverflow. Unfortunately the migration of the question is not possible after two months of posting. I have been reading about length spaces in the (great) book ...

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