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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Is every coarse map between proper geodesic spaces a quasi-isometric embedding?

Let $(S,d_{1})$ and $(S',d_{2})$ be two proper geodesic metric spaces. Is every coarse map $f: S\rightarrow S'$ a quasi-isometric embedding?. Just to recall, a coarse map $f: S\rightarrow S'$ between ...
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K-convexity, duality and in Banach Spaces

I am interested in K-convexity as defined in https://webusers.imj-prg.fr/~bernard.maurey/articles/typandco.pdf Page 24 bottom of page. I am struggling to show that K-convexity passes to the dual given ...
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Geodesic convexity of small balls in Alexandrov CBB spaces

If $X$ is a Riemannian manifold, it is known that, for $p\in X$, there is some $\epsilon>0$ such that $B(p,\epsilon)$ is geodesically convex. Geodesically convex means there is a minimizing ...
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When does the union of all geodesics equal the metric interval?

Definitions: Throughout let $(M,d)$ be a geodesic metric space [cf. p. 104 of EoD]* with $d$ the (strictly) intrinsic metric, i.e. for all $x,y \in M$, $d(x,y)$ equals the length of any minimizing ...
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Spectral radius of a finitely generated group

Let $G$ be a finitely generated group and $\Gamma$ be its Cayley graph with the usual word metric. Let $\mu$ be a non-degenerate measure on $G$, and define $p(x, y) = \mu(x^{-1} y)$. As is well-known, ...
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Proof that the sum of the exinradius is equal to 4 times the circunradius and the exinradius.

I have seen a problem that uses this identity in his solution however I haven't seen anywhere a proof of this lemma. I have read that is called steiner relation. I think that maybe carnot's theorem ...
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Algebraic Characterisation of the End Space of a proper geodesic space in terms of non-continuous functions

Let $X$ be a rimcompact Tychonoff space, that is, a completely regular Hausdorff space with a base of open sets with compact boundaries. It is well known that $X$ has a maximal compactification with a ...
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What is the degeneracy of the "Coxeter plane" for the E8 lattice?

The 240 minimal vectors (roots) of the E8 lattice, projected onto "the" Coxeter plane, are shown here: https://en.wikipedia.org/wiki/E8_(mathematics)#/media/File:E8Petrie.svg and discussed ...
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Union and intersection of tough domains

Imagine $M \subset \mathbb{R}^d$ that could be represented as a finite union of closures of pairwise disjoint domains $M_j$$(j=1, \cdots, n)$ Consider $S : M \to M$ such that all restrictions $S|_{(...
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Two points inside a triangle x distance away from each other, at least one of it must be x distance away from one of the vertices of a triangle.

Given a triangle $\triangle $ ABC. Suppose there are two points inside the triangle, and they are $p$ and $q$. Let $d(p,q)=x$, here d represents distance. How to prove this: if $\min\{d(A,p), d(C,q)\...
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How to show the limit space is totally bounded in the proof of completeness of Gromov-Hausdorff metric space?

The Gromov-Hausdorff metric space $(\mathcal{M},d_{GH})$ is complete. I'm currently following the proof of this fact given in Petersen's Riemannian Geometry (3rd Edition) (see Proposition 11.1.8). ...
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Is my proof that a constant speed geodesic is a geodesic correct?

Given a metric space $(X,d)$ the following is a definition of a geodesic from the book of Santambrogio. I want to show that a constant speed geodesic is a geodesic : is the following enough? Note ...
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Asymptotic bi-infinite geodesics in CAT($-1$) space coincides

Suppose $\gamma_1,\gamma_2:(-\infty,\infty)\to (X,d)$ be two bi-infinite geodesic such that $d(\gamma_1(t),\gamma_2(t))<K$ for all $t$. Here $(X,d)$ is a CAT$(-1)$ space. Then Image($\gamma_1$)=...
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Existence of a geodesic with endpoints at infinity

I want to prove the existence of a geodesic $\gamma$ with end points $\xi$ and $\xi'$ at infinity in a proper CAT(-1) space. A hint in Elements of Asymptotic Geometry is to first notice that each ...
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Derivative of distance along a smooth curve

I am struggling to solve the following problem from 'Introduction to Riemannian Manifolds' by John M. Lee $(M,g)$ be a connected Riemannian manifold. $\gamma:(-\epsilon,\epsilon)\to M$ be a smooth ...
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Quasi-isometry between an once-punctured torus and the Loch Ness Monster under suitable metrics

I was wondering if there are quasi-isometric (complete) metrics on the once-punctured torus and the Loch Ness Monster. By an once-punctured torus I mean the 1-genus surface with one puncture and by ...
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Floyd Compactification

Suppose $G$ is a finitely generated group, $f: \mathbb{N} \longrightarrow \mathbb{R_+}$ is a function satisfying $1<\frac{f(n)}{f(n+1)}<\lambda$ and $\sum f(n)<\infty$, which we called a ...
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Gromov product on the boundary

I was reading the book Elements of Asymptotic Geometry by Sergei Buyalo and Viktor Schroeder. There in the second chapter Gromov boundary is defined as the equivalence classes of sequences (in the ...
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Borel quasi-isometry between proper geodesic spaces

Let $(S,d_{1})$ and $(S',d_{2})$ be two proper geodesic metric spaces. If there exists a quasi-isometric embedding $f: S\rightarrow S'$, does there exist a $\textbf{Borel}$ quasi-isometric embedding $...
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Continuous quasi-isometry between Riemannian manifolds

Let $(S,d_{1})$ and $(S',d_{2})$ be two proper geodesic metric spaces. If there exists a quasi-isometric embedding $f\colon S\rightarrow S'$, does there exist a $\textbf{continuous}$ quasi-isometric ...
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Total Curvature for a curve in a metric space

Is there a theorem involving total curvature and some notion of index number for a curve in a metric space, as there is for planar curves? (i.e. total curvature is an integer multiple of $2\pi$.) I'm ...
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A set of points is contained in a sphere $S$. When is $S$ also the circumsphere?

Given points $p_1,\ldots,p_n\in\Bbb R^d$ so that all of them are contained in a common sphere $S\subset\Bbb R^d$ (by which I mean the usual $(d-1)$-dimensional sub-manifold of $\Bbb R^d$). Note that $...
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Tangent Cones at Infinity

A possible definition for a tangent cone at infinity is that a complete noncompact metric space with infinite diameter $(X, d_X)$ has a tangent cone at infinity $(Y, d_Y )$ if there is a sequence of ...
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Why aren't derivatives defined on metric spaces?

I'm studying the book by Ambrosio on gradient flows in metric spaces. It's stated that the usual notion of gradient flow, $$ x'(t) = -\nabla_x f(x),$$ is not defined on metric spaces because we don't ...
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Does this property of projection on metric space hold or not?

Let $(\mathcal{X}, d)$ be a complete metric space. For any subset set $C$ of $\mathcal{X}$, we say that $C$ is convex if for any two points $x, y, \in C$, there is a geodesic $\gamma (t)$ from $x$ to $...
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Does the Besicovitch Covering Property hold on finitely generated groups of polynomial growth when equipped with the word length metric?

Let $G$ be a finitely generated group with symmetric generating set $S$. Then $S$ induces a distance $d$ on $G$ by letting $d(a,b) = $ the minimum $n$ such that there are generators $s_1,...,s_n$ with ...
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If an isometry $f$ of a length space $X$ fixes a ball $B(x,r)$ for some $x$ and $r>0$, is it true that $f$ is the identity map?

Let $(X,d,L)$ be a length space, that is, $d = \inf L$, where the inf is taken over all curves with fixed endpoints. In $\mathbb{R}^2$ we have a results that says: if an isometry fixes 3 points, then ...
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Distance between geodesic rays 2

This is a follow up to my previous question which turned out to be wrong. Now my question instead is this: Given a $\delta$-hyperbolic space $(X,d)$ (with Rips triangle condition: a point on any side ...
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Distance between geodesic rays

I am trying to prove the following but so far did not succeed. Given a $\delta$-hyperbolic space $(X,d)$ (with Rips triangle condition: a point on any side is contained in the $\delta$-neighbourhood ...
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Maximum value of two sides of the length of a triangle in a circle, when two points of a triangle are fixed. [closed]

For three points $a,b,c$ on a circle, what is the maximum value of $ab+ac$ when points $b$ and $c$ are fixed. I believe that the maximum value of $ab+ac$ is when $a$ is in the middle of the bigger ...
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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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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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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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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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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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2 votes
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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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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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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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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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2 votes
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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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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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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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3 votes
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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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2 votes
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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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4 votes
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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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1 vote
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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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