# Questions tagged [mg.metric-geometry]

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

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### On tiling the plane with non-congruent, equal area triangles with each edge having a unique length

Ref: Tiling with incommensurate triangles shows an approach for tiling with incommensurate triangles - all sides and all angles unique and also with different areas - with the perimeters of the tiles ...

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### Trigonometry/spherical angles/minimum-least-squares

An issue from 3D tessellated geometry: Find the direction vector of a plane that minimizes the silhouette of a set of triangles. To say it another way, find the direction vector that is most ...

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### Do the heights of an acute triangle intersect at a single point (in neutral geometry)?

A well-known result of the Euclidean planimetry says that the heights of any triangle have a common point called the orthocentre of the triangle. This result is not true in neutral geometry (i.e., ...

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### Kissing behavior of planar regions

This post reworks a question that was stated in a slightly different form at Convex region $C$ with least kissing number of copies of $C$.
Background: Given a 2D region $C$ (not necessarily convex), ...

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### Finding angle with geometric approach [closed]

I would like to solve the problem in this picture:
with just an elementary geometric approach. I already solved with trigonometry, e.g. using the Bretschneider formula, finding that the angle $ x = ...

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### Uniformly continuous homotopy equivalence

Suppose $M$ and $N$ are complete metric spaces and $f, g: M \to N$ are uniformly continuous maps between them with common modulus of continuity $m$. Further suppose $f$ and $g$ are homotopy equivalent....

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### The closest ellipse to a given triangle

Definition: The Hausdorff distance between two point sets is the greatest of all the distances from a point in one set to the closest point in the other set.
Question: Given a general triangle T, to ...

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### Is the hypotenuse operation associative in every Tarski plane?

By a Tarski space I understand a mathematical structure $(X,\mathsf B,\equiv)$ consisting of set $X$, a betweenness relation $\mathsf B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\...

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### What does mean módulo 2pi? [migrated]

I was reading a paper and it have a equation inside absolute value with a small 2pi on the right corner , the paper explains
|.|2pi denotes modulo 2pi

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### Is an equilateral triangle constructible in a Tarski plane?

By a Tarski space I understand a mathematical structure $(X,\mathsf B,\equiv)$ consisting of set $X$, a betweenness relation $\mathsf B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\...

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### Concentration of volume towards the boundary

Consider a Euclidean space $X$ of large dimension $N$. For a measurable subset $G\subseteq X$ and $\varepsilon>0$ let
$$G_\varepsilon:=\{x\in G\mid B_\varepsilon(x)\subseteq G\}$$
be the set of all ...

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### On 'axiality' of planar convex regions

Axiality has been studied under a definition given here: https://en.wikipedia.org/wiki/Axiality_(geometry)
Consider an alternative definition of axiality as follows: For a convex region C, consider a ...

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### Does every Tarski plane embed into a 3-dimensional Tarski space?

By a Tarski space I understand a mathematical structure $(X,B,\equiv)$ consisting of set $X$, a betweenness relation $B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\times X^2$ ...

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### Maximal number of times distance $1$ can occur among $n$ points in the plane [duplicate]

For $n\in\mathbb N$, let $f(n)$ be the maximal number of times distance $1$ can occur among $n$ points in the plane:
$$
f(n) = \max_{ \{ x_1,\ldots,x_n \} \subset \mathbb R^2} \# \big \{ i<j : \| ...

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### When is Euclidean distortion finitely determined?

The Euclidean distortion of a metric space $X$, denoted $c_2(X)$, is the infimum of $c$ for which there exists a map $f\colon X\to\ell^2$ such that
$$d_X(x,y) \leq \|f(x)-f(y)\|_{\ell^2} \leq c\cdot ...

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### Smallest centrally symmetric containers of planar regions - 2

This post adds a bit to Finding the smallest centrally symmetric region that contains a convex planar region . In his answer there, Jukka Kohonen observed: Given a planar convex region $C$, the ...

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### Minimum diameter of set inscribed in a unit sphere

For a study of the stability of certain maps taking values in a sphere I have the following question.
Let $A$ be a subset of $\mathbb{R}^n$. Suppose $A$ lies in a unit ball, but in no ball of smaller ...

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### What is the area-decreasing 'convex hull'?

Let $K \subset \mathbf{R}^3$ be a compact set.
What is the smallest set $C$ containing $K$, with the property that in a neighbourhood of $C$, the closest-point projection of surfaces onto $C$ ...

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### Does every ‘curvature’ tensor induce a metric? [duplicate]

So we know that given a Riemannian manifold $(M,g)$ we can calculate its Riemannian curvature $R$. This covariant $4$-tensor field then satisfies some important symmetries
\begin{gather*}
R_{ijkl} = - ...

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### Computability of fillability of unit cube in $\mathbb{R}^n$ by $k$ $\varepsilon$-balls

Let $\mathbb{N}$ denote the set of positive integers. We define a relation $R \subseteq \mathbb{N}^4$ in the following way:
$(p,q,n,s)\in R$ if and only if there is $S\subseteq [0,1]^n$ with $|S| = s$...

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### Is a right triangle with the given cathetus and the opposite angle constructible in the absolute plane?

Question. Is it possible to construct a right triangle with a given cathetus $a$ and a given opposite angle $\alpha$ using only compass and ruler in the absolute geometry (so without the axiom of ...

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### de Rham's trisection method - English

I want to learn more about de Rham's trisection method in
De Rham, Georges, Un peu de mathématiques à propos d'une courbe plane, Elemente der Mathematik 2 (1947): 73-76. http://eudml.org/doc/140463.
...

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### Reference for Varopoulos isoperimetric inequality with multiplicity

The Varopoulos isoperimetric inequality for a bounded domain $D$ in a nilpotent group $\Gamma$ of growth $n$ reads
$$
\# D \le \mathrm{const} \cdot (\#\partial D)^{n/(n-1)}
$$
See Ch. 6.E+ in Gromov's ...

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### How to determine if a finite vertex split exists for a triangle mesh patch?

Setup: Suppose I'm given an ordered sequence of $n\ge 3$ points in 3D with
$$\vec{p}_i \in \mathbb{R}^3 \text{ for } i \in [1,n]$$ and some special selected index $k \in [2,n]$.
I'm also assured that ...

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### Can a billiard rack be a square, for every number of balls?

A billiard rack is a rack, usually a triangle, that can hold a certain number of equal size billiard balls, such that the balls' centres cannot move within the rack.
Can the rack be a square, for ...

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### Is the centroid property equivalent to the middle line property of the triangle?

By a Tarski plane I understand a set $X$ endowed with a betweenness relation $\mathsf B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\times X^2$ satisfying all Tarski axioms except ...

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### Metric space whose bounded subsets are totally bounded

Is there a name for a metric space in which any bounded subset is totally bounded, or equivalently, in which any bounded sequence contains a Cauchy subsequence?
I have seen the name Bolzano-Weierstraß ...

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### Proof of Lemma 37.5 in Pak's Lectures on Discrete and Polyhedral Geometry

I am staring at the proof of Lemma 37.5 in [Lectures on Discrete and Polyhedral Geometry][1], see page 331.
I cannot understand why the required triangulation exists.
In the first paragraph it says &...

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### Bisector of two points in a Riemannian manifold has measure $0$

Let $p,q\in M$, $p\neq q$, where $M$ is a Riemannian manifold. We will let the bisector of $p,q$ be $\mathcal{B}(p,q)=\{x\in M;d(p,x)=d(q,x)\}$. Does $\mathcal{B}(p,q)$ have measure $0$?
I was ...

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### Do cut-length-minimizing equidissections exist?

Suppose $A,B$ are polygons of equal area. By the Wallace-Bolyai-Gerwien theorem, $A$ and $B$ are equidissectable: we can make finitely many straight-line cuts in $A$ and rearrange the resulting pieces ...

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### Constrained morphing of polygons

This post continues 'Constrained morphing' of planar convex regions
If an $m$-gon $P_m$ is to be morphed (altered continuously) into an $n$-gon $P_n$ with same area and perimeter, can one ...

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### Uniformly open map on a dense subset

Schauder's lemma asserts that you can always extend a uniformly continuous, uniformly open map from a dense subset of a complete metric space to a uniformly open map on the completion.
I think the ...

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### 'Constrained morphing' of planar convex regions

Morphing may be defined as a continuous transition of one shape to another. This post is about modifying planar regions continuously from one form to another under some constraints.
Qn: If $C_1$ and $...

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### Dividing a spherical cap into $n$ equal wedges

This is a follow-up of the question Dividing a spherical cap into three equal wedges where the $n=3$ case was shown.
Motivation: Optimal ways to cut an orange.
In this problem, we have a spherical ...

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### Is there a 5-cell-600-cell honeycomb?

Is there a convex uniform tiling of hyperbolic 4-space with 5-cells and 600-cells as its facets and a snub 24-cell as its vertex figure?

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### Possible extensions of the perpendicular axes theorem for moment of inertia

This post continues on Moment of inertia from Bisectors and partitioning lines for convex regions defined with respect to the moment of inertia.
The perpendicular axis theorem states that the moment ...

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### On diffeomorphisms that preserve the metric

Suppose $\Omega\subset \mathbb R^2$ is a bounded domain with smooth boundary and suppose that
$$ F: \Omega \to \Omega,$$
is a diffeomorphism that fixes $\partial \Omega$ (i.e $F|_{\partial \Omega}$ is ...

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### Hausdorff dimension of the non-differentiability set of a locally Lipschitz function

Let $f:\mathbb R^n \to \mathbb R$ and $E := \{x \in X : f \text{ not Fréchet differentiable at }x\}$. Then $E$ is Borel measurable. It is well-known that
Theorem If $f$ is convex, then the Hausdorff ...

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### Lipschitz maps with Hölder inverse preserve the doubling property

Let $K$ be a compact doubling metric space, $X$ be a metric space and $f:K\rightarrow X$ be Lipschitz with $\alpha$-Hölder inverse, where $0<\alpha<1$. Does $f(K)$ need to be doubling?

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### A topological tree is weakly contractible

Let us call a nonempty topological space a topological tree if it is Hausdorff and for two distinct points there is a continuous injective path connecting the points, which is unique up to ...

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### Regular $n$-gon with diagonals: bounds on area of largest cell?

Consider a regular $n$-gon of side length $1$ with diagonals. Here is an example with $n=11$ (from geogebra applet).
I've been trying to find, in terms of $n$, bounds on the area of the largest cell, ...

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### Condition to be concyclic [closed]

What condition would you impose upon $n$ points on a plane of which no three points are collinear so that they are
concyclic if the distances of each point from the all remaining points are known? (...

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### Comparing the areas of polygons via equidecomposability in the hyperbolic plane

It is well-known that in the Euclidean plane two simple polygons have the same area if and only if they are equidecomposable, i.e., can be decomposed into congruent triangles.
Question. Is an ...

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### On local-to-global theorem of $\mathrm{CD}^*(K,N)$ spaces

In the 2010 JFA paper "Localization and tensorization properties of the curvature-dimension condition for metric measure spaces" (arXiv link, DOI link), the authors used Theorem 5.1 to prove ...

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### Local Lipschitz constant of exponential map for Hadamard manifolds

Suppose that $(M,X)$ is a simply connected complete Riemannian manifold with pinched sectional curvature between $[a,0]$. Let $r>0$ and fix any point $p\in M$. Is there a bound on the local ...

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### Why is the normal cone of a face of a polytope orthogonal to that face

I understand the definition of normal cone and normal fan as it relates to polytopes, but I'm having a hard time proving the well known fact that the normal cone of a face of my polytope is orthogonal ...

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### How much of an aperiodic tiling is needed to force aperiodicity?

Consider an aperiodic tiling. By definition, there is a $C$ such that, for any box of side $C$, the part of the tiling contained in the box can be continued to the whole plane only in a non-periodic ...

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### How many distances are required to calculate all distances among $n$ points in the Euclidean plane?

I want to know all the pairwise distances between points $P_1,P_2,\ldots,P_n$ in the Euclidean plane (or equivalently, I want to reconstruct the set of points up to congruence). Let's say I have an ...

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151
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### Covering unit-radius balls with unit-diameter objects

Let $d$ be a norm-based metric in $\mathbb{R}^2$. We are given a $d$-ball with radius 1, and we would like to cover it with objects with diameter 1. How many objects are needed?
In the $\ell_1$ metric,...

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### What is the value of the (p,1)-summing norm of the Euclidean plane for p=2?

The general definition of the $(p,q)$-summing norm
of Banach space $X$ is
$$
\pi_{(p,q)}(X) :=\inf \bigg\{\gamma\in\mathbb{R}: \forall n\in\mathbb{N},\; x_1, \ldots, x_n \in X: \notag\\ \qquad\...