# 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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1 vote
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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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### 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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### 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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1 vote
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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 ...
1 vote
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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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### 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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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\...