# Questions tagged [mg.metric-geometry]

Euclidean, hyperbolic, discrete, convex, coarse geometry, comparisons in Riemannian geometry, symmetric spaces.

3,418
questions

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### What do you call this notion of similarity between subsets of a metric space?

Given a metric space $X$, we say a continuous isometry is a continuous map $F:X\times[0,1]\to X$ such that defining $f_t(x) = F(x,t)$ for $t \in [0,1]$, we have that $f_0$ is the identity map and $f_t$...

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### Minimize the z coordinate for the intersection of two extrusions of coplanar circles along parametric paths

Sorry if my question isn't phrased as formally as it should be; this is my first math question on any online forum.
I basically want to minimize the z coordinate of the intersection of some number of ...

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**1**answer

40 views

### intersection and self intersection on non compact complex surfaces

I'd like to have a definition of intersection on non compact complex surfaces because all i have found so far is only about projective surfaces. for example how can i define the self intersection of ...

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61 views

### Construct by compactness (Pentagonal tiling – Rao paper)

In the (arxiv) paper, Exhaustive search of convex pentagons which tile the plane, on page 4 under the proof of Lemma 2, it is said that:
"... We keep a connected component $H_d'$ of $H_{d}$ such ...

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395 views

### Which unfoldings of the $d$-dimensional hypercube tile $(d{-}1)$-space?

A six year old question,
Which unfoldings of the hypercube tile $3$-space?, has just been answered by
Moritz Firsching:
All $261$ unfoldings tile space!
So now we know:
For $d=2$, the unfolding of ...

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**1**answer

68 views

### Inequality in a triangle associated with Golden ratio

Let $ABC$ be arbitrary triangle, $D$, $E$, $F$ are the midpoints of $BC$, $CA$, $AB$ respectively. Define points, segments in the figure belows. I am looking for a proof that:
$$DE+EF+FD \le (DG+DH+...

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64 views

### When is semi-metric topology Hausdorff

Let $(X,d)$ be a semi-metric space (i.e.: $d$ is a metric on $X$ if it would satisfy the triangle inequality). My question is simple: Are there known "non-trivial" conditions under which ...

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### Condition for: A simple quotient metric induced by surjective map + equivalence relation

Let $X$ be a metric space and let $f:X\rightarrow Z$ be a surjective map onto some set $Z$. Define the pseudo-metric $d_f$ on $Z$ by:
$$
d_f(z_1,z_2)\triangleq \inf_{\underset{f(x_i)=z_i}{x_i\in X}}
\...

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68 views

### Gromov–Hausdorff closure of non-positively curved graphs

Setup:
Let $\Gamma$ be the set of non-positively curved weighted connected graphs, with finitely many points, which are isometrically embedded in $\mathbb{R}^n$; for some $n\in \mathbb{N}$;$n\geq 2$. ...

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80 views

### Covering number of smooth functions from $\mathbb{R}^d$ to $\mathbb{R}$

Let $(\mathcal{X},d)$ be a space of function $f: \mathbb{R}^d \to \mathbb{R}$ where $d=\| \cdot \|_\infty$ (i.e., $d(f)= \sup_{x\in \mathbb{R}^d} |f(x)|$ ).
Let $D_\alpha f= \frac{\partial^\alpha}{ \...

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### Cutting convex polygons into triangles of same diameter

This question continues from: Cutting convex regions into equal diameter and equal least width pieces
Definitions: The diameter of a convex region is the greatest distance between any pair of points ...

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76 views

### Is the Jaccard distance between probability vectors a metric?

Let X and Y be probability vectors, meaning that X = $[x_1, x_2, ..., x_n]^T$, where $x_i\leq 1$ and $\sum_{i=1}^{n}x_i=1$ (Y is defined similarly).
Define the Jaccard distance as
\begin{equation}
...

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561 views

### An introductory text on expanders

I am looking for a book that covers expander graphs rigorously. Preferably a book aimed at beginners.

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67 views

### Completeness on the tangent bundle

I was wondering if geodesics are defined for all time on compact Finsler manifolds, which I know very little about.
In an attempt to prove this, I thought maybe I could show that if $M$ is compact, ...

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45 views

### Equal products of triangle areas

Can you prove the following claim:
Claim. Given hexagon circumscribed about an ellipse. Let $A_1,A_2,A_3,A_4,A_5,A_6$ be the vertices of the hexagon and let $B$ be the intersection point of its ...

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294 views

### Golden ratio as a property of conic section (is it known?)

I am looking for a proof of a discovery as follows:
Let $ABC$ be arbitrary triangle and $(\Omega)$ be an arbitrary circumconic of $ABC$ let $A'B'C'$ is its tangential triangle of $ABC$ respect to $(\...

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236 views

### Thirteen-point conic and four-point line, are they new?

We know that Five points determine a conic and Two Points Determine a Line. Here I found a simple construct of a conic through $7$ points (in PS I note that how the conic through thirteen points) and ...

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**1**answer

189 views

### Maximizing the distance sum of some points inside a circle

Consider $n$ points $\{p_i\}_{i=1}^n$ located inside or on a circle with radius $r$ in the plane. The question is: how to place the $n$ points so that the sum of inter-point distances,
$$J=\sum_{i=1}^...

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554 views

### Is every 1-million-connected graph rigid in 3D?

It is an old result that every $6$-connected graph is rigid in $\mathbb{R}^2$:
Lovász, László, and Yechiam Yemini. "On generic rigidity in the plane." SIAM Journal on Algebraic Discrete ...

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493 views

+250

### Smoothness of distance function to a compact set

Fix a non-empty compact subset $K\subseteq \mathbb{R}^n$ and let $d_K(x):=\min_{z \in K} \,\|z-x\|$ be the map sending any $x\in \mathbb{R}^n$ to its distance from $K$.
Suppose that:
$K$ is regular : ...

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### Is it a new discovery on conic section?

I discovered a problem in plane geometry (there are some nice special cases) as follows:
Let $ABC$ be a triangle and $\Omega$ be arbitrary circumconic. Let two points $A_b, A_c \in BC$, $B_c, B_a \in ...

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### Does there exist a countable metric space which is Lipschitz universal for all countable metric spaces?

Is there a countable metric space $U$ such that any countable metric space is bi-Lipschitz equivalent to a subset of $U$? How about $c_{00}(\mathbb{Q})$ where $\mathbb{Q}$ is the rational numbers? ...

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81 views

### Generalizing Bottema's theorem

Can you provide another proof for the claim given below?
Claim. In any triangle $\triangle ABC$ construct triangles $\triangle ACE$ and $\triangle BDC$ on sides $AC$ and $BC$ such that $\frac{AE}{AC}...

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60 views

### What is the center of minimum distance of a region?

Suppose we have a compact plane region $R$ (not necessarily convex or connected). I am working in a problem which involves the point $p$ in $R$ that is, in average, the closest to every other point. ...

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216 views

### Facility location on manifolds

Facility location studies optimal placement of a certain number $n$ of points (facilities) in some region $R$. (https://en.wikipedia.org/wiki/Facility_location_problem)
The minimax facility location ...

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**1**answer

77 views

### Equal sums of line segments

I would like to see a proof of the following
Claim. Let $A_1,A_2,A_3,A_4,A_5$ be vertices of bicentric pentagon. Let $B_1$ be the intersection point of $A_1A_3$ and $A_2A_5$, $B_2$ the intersection ...

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321 views

### Topological dimension, Hausdorff dimension, and Lipschitz mappings

I can prove the following result. Here $\operatorname{dim} X$ stands for the topological dimension and $\mathcal{H}^n$ denotes the Hausdorff measure.
Theorem. Suppose that $f:\mathbb{R}^n\supset\...

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150 views

### Stability of isoperimetric inequality

Let $S$ be subset of $\mathbb{R}^n$ with perimeter 1.
Isoperimetric inequality states that then the volume of $S$ is not greater than $V_n$,
where $V_n$ is the volume of a ball in $\mathbb{R}^n$ with ...

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**3**answers

222 views

### Recovering the length metric from Hausdorff measure

The metric cannot be recovered from its Hausdorff measure in general. Now, assume that $(X,d_X)$ and $(Y, d_Y)$ are connected compact length spaces and induce $n$-dimensional Hausdorff measures $\...

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120 views

### Covering the sphere with an approximately planar grid

Consider a triangulation of a radius $R$ sphere into $n$ triangles. Must $Ω(\sqrt n)$ triangles have $Ω(1)$ relative difference from being an equilateral triangle of area $4πR^2/n$? ($Ω$ is from ...

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428 views

### Which knots appear as the singular locus of a polyhedral metric on the 3-sphere?

What can be said about a knot $K\subseteq S^3$ for which there exists a (Euclidean) polyhedral metric (aka Euclidean cone-manifold structure) on $S^3$ whose singular locus is precisely $K$? I'm ...

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200 views

### Are two quasi-isometric, isomorphic on large enough balls, transitive graphs isomorphic?

Take two transitive graphs $X,Y$ (potentially directed and edge-labelled, e.g. Cayley graphs).
Assume $X,Y$ are quasi-isometric with constant $K$, i.e. there exists a function $f:VX \to VY$ ($VX,\,VY$ ...

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64 views

### A criterion for loxodromicity in Gromov-hyperbolic spaces

Recall that an isometry of a Gromov-hyperbolic space $X$ is called loxodromic if it has exactly two fixed points on the Gromov boundary $\partial X$, one being "attracting" and the other &...

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### A generic question on circles associated with a triangle

This question is inspired by two posed by P.Terzić (both given elegant synthetic proofs by F. Petrov). The starting point is a triangle $ABC$ and a triangle centre $G_1$. There are two classical ...

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212 views

### Necessary and sufficient condition for quadrilateral to be cyclic

Can you provide a proof for the following proposition:
Proposition. Given any quadrilateral $ABCD$. Let $P,Q,R,S$ be nine-point centers of triangles $\triangle ABD$,$\triangle ABC$,$\triangle BCD$ ...

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42 views

### Explicit calculation algorithm for distance function [duplicate]

I study differential geometry. Although there is a lot of study on the local theory, a global description lacks some explicit explanations. I mean, the study of surfaces describes curves, tangent ...

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525 views

### Acute triangles in “obtuse” polygons?

Let $P$ be a convex polygon. Suppose every interior angle of $P$ is obtuse. Is it always the case that there exist three vertices $p, q, r$ of $P$ such that $\triangle pqr$ is acute?
I conjecture ...

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### Completeness of intrinsication

Lemma. Suppose $(X,\rho)$ is a complete metric space and $\hat \rho$ is its induced intrinsic metric. Then $(X,\hat \rho)$ is complete.
This lemma was essentially proved in [2.3. in Metric minimizing ...

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### Three circles intersecting at one point

Can you provide a proof for the following proposition:
Proposition. Let $\triangle ABC$ be an arbitrary triangle with nine-point center $N$ and circumcenter $O$. Let $A',B',C'$ be a reflection points ...

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**1**answer

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### Collinearity of three significant points of bicentric pentagon

Can you provide a proof for the following claim?
Claim. Given bicentric pentagon. Consider the triangle whose sides are two diagonals drawn from the same vertex and side of pentagon opposite from ...

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### Completion of an Alexandrov space

Let $X$ be an incomplete Alexandrov space with sec $\ge -1$ in the sense that for any point in $X$ there exists a small neighborhood in which the four-points criterion is satisfied.
Suppose $X$ is ...

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### Asymmetry of projections

A possible measure of asymmetry for a convex body $K \subset \mathbb{R}^n$ is the affine-invariant
quantity
$$
\alpha_n(K) := \frac{\textrm{vol}(K - K)}{2^n\textrm{vol}(K)} .
$$
Indeed, the Brunn-...

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### Does anyone know of an academic reference for this proof that the tangent to a hyperbola at a point bisects the angle between each focus to the point?

I am writing a report and as part of it I need to prove the property that for a point $P$ on a hyperbola, the tangent to the hyperbola at $P$ bisects the angle $\angle F_1PF_2$ where $F_1$ and $F_2$ ...

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133 views

### Does there exist an isometry between a regular polygon and a circle?

In order to define the question in a meaningful fashion, I am referring to a smooth manifold $\mathcal{M}$ within an $\epsilon$-neighborhood of a regular polygon $\mathcal{P}$ satisfying $$\max\{\|x-p\...

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139 views

### Adjunction formula for non compact surfaces

Let $M$ be a non compact complex surface and S an embedded compact Riemann surface in $M$.
I already know how to show the following equality of fiber bundle:
$$\Omega^2_{M}|S =\Omega^1_S \otimes N^*_S$...

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### Are there examples of hyperbolic manifolds with finite Bowen-Margulis measure and fundamental group which is not relatively hyperbolic?

It is well known that a geometrically finite hyperbolic manifold (quotient of $H^n$) has finite Bowen-Margulis measure.
Marc Peigné [1] constructed examples of geometrically infinite hyperbolic ...

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54 views

### Distance Metric on a Polytope

Primary Question: Is it possible to define a distance metric on a polytope (or permutohedron in particular)? I am aware that neither is a smooth, Riemannian manifold; however, computer scientists have ...

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62 views

### A generalized norm function in $\mathbb{R}^n$ [closed]

We defined a new norm. The norm of $x \in \mathbb{R}^n$ is defined as
$$ N_P(x) = \min \{t \geq 0 : x \in t\cdot P\} \enspace,$$
where $P$ is a centrally symmetric and convex body centered at the ...

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118 views

### Least square assignment and hyperplanes

Let $S$ be a finite set of points in $\mathbb{R}^{d}$, $c(s) \in [0,1]$ such that $\sum_{s \in S} c(s) = 1$, $\rho$ continuous and non-vanishing probability distribution on $[0,1]^{d}$ and $\mu $ ...

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3k views

### Can the fugitive escape?

A fugitive is surrounded by $N$ police officers, with the nearest one at distance $1$ away. The fugitive and the officers move alternatively.
In a fugitive move, the fugitive can travel no more than ...