# Questions tagged [mg.metric-geometry]

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

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### finding area of intersection of a rectangle and circle [closed]

I have a rectangle intersecting with a circle. I want to find the area of intersection shown in blue in the figure. I am calculating the area of rectangle as: A = b x (50-d) would that be the ...
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### Reference for “every 5-dimensional polytope has a 3-gonal or 4-gonal face”

It seems to be folklore that every 5-dimensional convex polytope has a 3-gonal or 4-gonal face of dimension two. I was not able to track down a source for that claim. Alternatively, I would be ...
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### An arrangement of entangled squares

Is there an arrangement of finitely many axes-parallel squares in the plane, of $k$ different colors, such that: The squares of each color are pairwise-disjoint; Each square overlaps at least $4$ ...
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### Is there a 4-polytope without 3-gonal and 4-gonal faces, other than the 120-cell?

The question is in the title: Question: Is there any 4-dimensional polytope without 3-gonal and 4-gonal faces (of dimension two), other than the 120-cell? I consider only convex polytopes (convex ...
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### Smoothing continuous functions in metric space

Let $(X,\rho)$ be a metric space. For any $f:X\to\mathbb{R}$, define the local Lipschitz constant of $f$ at $x$ by $$\Lambda_f(x) := \sup_{x'\in X\setminus\{x\}} \frac{|f(x)-f(x')|}{\rho(x,x,')} .$$...
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### Probability that a Voronoi cell contains exactly k random points

Gilbert's argument  for a given point to be contained in a Voronoi cell of area $s$ is that, known the p.d.f. of cell areas -- be it $f(s)$ --, then the probability of $f(s|X=1)=sE[s]^{-1}f(s)$ ...
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### Quotient distance on $\mathbb{C}P^n$ is equivalent to distance induced by the Fubini-Study metric

Let $n$ be an even, positive integer. Then, is the metric (in the metric-space sense) on $\mathbb{C}P^n$ induced by the Fubini-Study (Riemannian) metric equivalent to the quotient (pseudo?)-metric ...
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### Length and curvature for closed curves in negatively curved spaces

In the Euclidean plane, for a closed smooth curve of length $\ell$ whose curvature is bounded above by $\epsilon$ we have the inequality $$\ell \ge 2\pi \epsilon^{-1}$$ which follows from the fact ...
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### Properties of shapes defined by locus of points with a function of distances

Different shapes such as hyperbola and ellipse can be defined as a locus of points. For example, if we denote distance to points $P_1$ and $P_2$ from any arbitrary point as $d_1(x,y)$ and $d_2(x,y)$. ...
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### Sizes of connected components from a random choice in a grid

This is inspired by the illustration in this recently updated question. So we take a (fairly big) $n$ and an $n \times n$ grid where we draw at random one diagonal in each of the $1 \times 1$ squares. ...
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### What does the extension theorem for tilings state?

I have seen several references to the so-called Extension Theorem in the context of tilings of Euclidean space. E.g. in "The Local Theorem for Monotypic Tilings" one reads The Extension Theorem [......
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### Lebesgue measure of set of equidistant points with respect to a finite set

Let $X$ be a finite subset of $\mathbb{R}^n$; equip $d$ with a metric, and let $\emptyset \subset X\subseteq \mathbb{R}^n$ be of cardinality $N>0$. What requirements on my metric do I need so ...
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### Domains in $\mathbb{R}^n$ for which Hajlasz-Sobolev spaces and Sobolev Spaces are the same

I'm reading Heinonen's book on metric measure spaces. He writes that for general domains $\Omega \subset \mathbb{R}^n$, $M^{1,p}(\Omega) \subset W^{1,p}(\Omega)$ where the former are Hajlasz-Sobolev ...
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### $\varepsilon$-net of a $d$-dimensional unit ball formed by power set of $V = \{+1, 0 -1\}^d$

I have a set of $d$-dimensional vectors $V = \{+1, 0, -1\}^d$. Then $P(V)$ constitutes the power set of $V$. I now construct a set of unit vectors $V_{\mathrm{sum}}$ from the power set $P(V)$ such ...
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### Unknown work of Nöbeling on topological/Hausdorff dimension

Let $\mathcal{H}^n$ denote the Hausdorff measure, $\dim_H X$ the Hausdorff dimension, and $\dim X$ the topological dimension of $X$. A well known result of Szpilrajn (He changed his name to ...
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### Hausdorff metric selectors

Let $\ M\$ be the family of all non-empty bounded regular open subsets of $\ \Bbb R,\$ where regular means that every $\ G\in M\$ is equal to the interior of its closure. Let distance $\ d(G\ H)\$...
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### Motivation for persistent homology with respect to eigenfunctions of distance kernel operator in a recent preprint

I have a question about a recent preprint https://arxiv.org/pdf/1912.02225.pdf by Maria, Oudot, and Solomon. As far as I understand, in Section 8 they prove that persistent homology (persistence ...
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### Bound between distance between Rotation Matrices

Let $\|\cdot\|_F$ denote the Fröbenius norm on the set of $d\times d$ matrices. By restriction this induces a metric on $SO(n)$. Let's make an observation. Since $X\in SO(n)$ is a rotation matrix ...
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### What do you call $\operatorname{diam} (A)^d/\mathcal{L}^d (A)$ for $A \subseteq \mathbb{R}^d$ convex?

If $A \subseteq \mathbb{R}^d$ is convex, is there a more or less established name for the quantity $$\operatorname{diam} (A)^d/\mathcal{L}^d (A),$$ where $\mathcal{L}^d (A)$ is the Lebesgue measure of ...
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### Isometric stratification preserves volume?

Let $K\subset \mathbb{R}^k$ be a non-empty compact subset let $f:K \to K$ be Lipschitz and surjective. If, moreover, $f$ is an isometry then clearly $f$ preserves the Lebesgue measure of $K$. I ...
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### Which convex bodies can be captured in a knot?

Which convex bodies can be captured in a knot? This question is based on the discussion in "Is it possible to capture a sphere in a knot?". We assume that the knot is made from unstretchable, ...
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### Order-relational conditions for 4 points being in convex configuration

In the euclidean plane a simple sufficient condition for 4 points being in convex configuration is as follows: if the points are $\lbrace A,\,B,\,C,\,D\rbrace$ of which $\lbrace A,\,B,\,C\rbrace$ are ...
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### Gromov-Hausdorff distance between graphs with edges as part of the space versus not part of the space

Let $G_1$ and $G_2$ be finite simple graphs viewed as metric spaces in the natural way where the edges are not part of the space. Let $G_1'$ and $G_2'$ be copies of $G_1$ and $G_2$ resp. but with the ...
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### Area of an intersection of three ellipses

Let $\Delta := (A,B,C)$ be a triangle that is defined by three points in the Euclidean plane that are not collinear. Let further $E_{(A,B),\,C},\,E_{(C,A),\,B},\,E_{(B,A),\,B}$ be the set of ellipses ...
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### Metrics on the space of distributions in terms of p.d.fs

If two probability distributions (on the same measure space) are s.t they have p.d.fs and the $L^1$ distance between the p.d.f.s is large, then is there a choice of a nice" metric $d_{\rm smooth}$ ...
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### Which curves and surfaces are realizable by linkages? references?

Ok, so I try to formulate rigorously the question in the title, for which I am asking for references. My definitions may be flawed, so feel free to adjust/correct them! I care about dimensions 2 and 3 ...
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### For which classes of metric spaces can we prove that quasi-isometry is an equivalence relation in ZF?

Given two metric spaces $(M_1, d_1)$ and $(M_2, d_2)$, a map $\phi \colon (M_1, d_1) \to (M_2, d_2)$ is a large-scale Lipschitz essentially surjective map if there exist constants $A \geq 1, B \geq 0$,...
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### Effective radius of section of a convex set compared to that of the convex itself

The effective radius $er(A)$ of a $n$-solid $A$, is defined by Schramm (see the question by Gil Kalai Volumes of Sets of Constant Width in High Dimensions) to be the radius of the $n$-ball that has ...
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### Injectivity of a locally strictly expanding map on a compact space

Prove that any locally strictly expanding map on an infinite compact metric space is non-injective.
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### Bounding the total variation metric between Gaussian mixtures

Let $\mathcal{P}(\mathbb{R}^d)$ the space of probability measures on $\mathbb{R}^d$ with total variation metric $\delta$, fix $k \in \mathbb{N}$, and let $\mathcal{P}'\subset \mathcal{P}(\mathbb{R}^d)$...
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