Questions tagged [mg.metric-geometry]

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

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21 views

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 ...
3
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1answer
75 views

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 ...
3
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2answers
85 views

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 $4$ ...
3
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1answer
128 views

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 ...
4
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0answers
65 views

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 $$ \Lambda_f(x) := \sup_{x'\in X\setminus\{x\}} \frac{|f(x)-f(x')|}{\rho(x,x,')} . $$...
3
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0answers
26 views

Probability that a Voronoi cell contains exactly k random points

Gilbert's argument [1962] 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 $f(s|X=1)=sE[s]^{-1}f(s)$ ...
6
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1answer
96 views

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 ...
6
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0answers
83 views

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 ...
2
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0answers
16 views

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 $d_2(x,y)$. ...
8
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1answer
85 views

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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1answer
55 views

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 [......
2
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0answers
47 views

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 ...
3
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0answers
35 views

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 ...
4
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1answer
107 views

$\varepsilon$-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 ...
6
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2answers
309 views

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 ...
2
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2answers
97 views

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)\ $...
3
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0answers
58 views

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 ...
3
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1answer
42 views

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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40 views

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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1answer
74 views

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 ...
7
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0answers
137 views

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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16 views

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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53 views

Polytopes with large dihedral angles

The regular $d$-simplex has dihedral angle $\arccos(1/d)<90^\circ$, and the $d$-cube has dihedral angle exactly $90^\circ$. The maximal dihedral angle of a prism over a $(d-1)$-simplex is also $90^\...
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0answers
73 views

Determining the behavior of a contraction mapping with undefined points

Label $X$ as the real interval $[0, a]$ where $a \in \mathbb{R}^+$, so that $\text{int}(X) = (0, a)$ labels the interior of $X$ and $\partial X$ labels the boundary of $X$. I have a function $f:\text{...
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0answers
38 views

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 ...
2
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0answers
71 views

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 ...
3
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0answers
141 views

Sets of finite perimeter: intersection with an half space

I have a question regarding sets of finite perimeter. In particular I'm interested to find $$\mu_{E \cap H_t}, \label{1}\tag{1}$$ where $E$ is a set of finite perimeter in a generic open set $\Omega \...
6
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1answer
85 views

Continuity/Lipschitz regularity of exponential map from $C_c$ to $\operatorname{Diff}_c$?

For finite-dimensional Lie algebras, see this for a nice example, the exponential map is smooth and in particular, it is locally-Lipschitz onto its image. However, things are different when moving to ...
7
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2answers
165 views

Maximal distance of $2d+1$ points on a sphere

If one arranges $2d$ points on the sphere $\mathbf S^{d-1}\subset\Bbb R^d$ as the vertices of the regular octahedron, then one can achieve a minimal spherical distance of $\pi/2$ between any two ...
1
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0answers
28 views

Associativity property of the gyrobarycenter

I'm using Ungar's terminology and notations. In the open unit ball of $\mathbb{R}^n$, let $GB(A_1, \ldots, A_N; m_1, \ldots, m_N)$ be the gyrobarycenter of the points $(A_1, \ldots, A_N)$ with ...
1
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2answers
193 views

A question about dense sets

Suppose that $A$ is a given subset of $I=[0,1],\ $ and $ \left\{ x_j = \frac{j}{m} \right\}_{j=0}^{m}\ $ is the $m$-partition of $I$, and $\nu(m)$ is the number of $\ [x_{i-1},x_{i}]\ $ such that $\ [...
0
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0answers
32 views

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 $d_{\rm smooth}$ ...
11
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2answers
557 views

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 ...
5
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0answers
122 views

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 $A \geq 1, B \geq 0$,...
3
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0answers
56 views

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 ...
0
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1answer
74 views

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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0answers
38 views

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 $\mathcal{P}'\subset \mathcal{P}(\mathbb{R}^d)$...
1
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0answers
25 views

A map similar to a foot map on a cap of a strictly convex surface

Consider a strictly convex surface $\Sigma$ in $\mathbb{R}^3$ homeomorphic to a sphere. When $p$ is a point not in a convex hull of $\Sigma$, then $\Sigma'$ is the boundary of convex hull of $p$ and $\...
8
votes
2answers
224 views

Graph metric approximating Euclidean metric

I've been reading Wolfram's recent articles about graph/mesh/grid structures as an analogy for physical space, and it seems to me that there will be a problem getting the notion of distance to work ...
1
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0answers
53 views

Can every quadrangle whose corners are in little squares be more square-like when we allow corners to move in the little squares [closed]

Let $f: P^4 \to \mathbb R^+$, $(A,B,C,D)\mapsto (AB-BC)^2+(BC-CD)^2+(CD-DA)^2+(AC-BD)^2$. Where $AB$ is the distance betwin $A$ and $B$. Let $A$, $B$, $C$, $D$ be four points of the plane $P$, ...
0
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1answer
93 views

Lower bound for mutual inner products of N random unit vectors in $\mathbb{R}^n$, N > n

I have $N$ independent random unit vectors $\{v_i\}$ in $\mathbb{R}^n$, where N > n. I need a concentration inequality of the form $$\text{P}(|v_i \cdot v_j| > \epsilon \,\,\,\, \forall i, j = 1, \...
2
votes
2answers
97 views

Is the point giving the width in strictly convex surface a cut point?

Assume that $\Sigma$ is a stricly convex surface in $\mathbb{E}^3$ homeomorphic to a sphere. Further, assume that $p_0,\ p_1\in \Sigma$ are intersection points with planes $z=0,\ z=1$ and the surface $...
5
votes
1answer
122 views

Orientations of triples of points in the plane

Given a finite indexing-set $I$ and a collection $P = \{P_i: \ i \in I\}$ of points in the plane no three of which are collinear, let $I_{(3)}$ denote the set of ordered triples of distinct elements ...
20
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2answers
1k views

Intercept the missile

A stealth missile $M$ is launched from space station. You, at another space station far away, are trusted with the mission of intercepting $M$ using a single cruise missile $C$ at your disposal . ...
3
votes
0answers
64 views

The divergence function and quasigeodesics in $\delta$-hyperbolic spaces

(From Bridson and Haefliger's Metric Spaces of Non-positive Curvature) Let $X$ be a metric space. A map $e: \mathbb{N} \rightarrow \mathbb{R}$ is said to be a divergence function for $X$ if the ...
5
votes
1answer
173 views

Quasi-isometric embedding of graphs in non-compact riemannian surfaces

Given a complete riemannian surface $(S,m)$, where $S$ is homeomorphic to $\mathbb{R}^2$, I would like to find a weighted graph $G$ (which means a graph with real non-negative weights on the edges), ...
2
votes
0answers
37 views

Chord of fixed length traveling around a Jordan curve

Let $C$ be a Jordan curve with nice enough properties whenever necessary (e.g. smooth, or just rectifiable, perhaps). I am interested in knowing how long can a chord be that "traverses" the Jordan ...
2
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0answers
120 views

Optimization with parametric constraints: solution maps

For constrained optimization problems $$ \begin{array}{ll} \min\limits_{x \in \mathbb R^n} & f(p, x) \\ \text{s.t.} & x \in C \end{array} $$ where $p \in \mathbb R$ is a parameter, we can ...
6
votes
1answer
199 views

Minimum area of the convex hull of the union of a parallelogram and a triangle

This question is somewhat dual to my previously stated question about Maximum area of the intersection of a parallelogram and a triangle, where the triangle and parallelogram each is assumed to be of ...
1
vote
1answer
296 views

coordinate free foundations of trigonometry [closed]

What axioms for geometry and trigonometry would I have to chose in order to completely avoid coordinates in defining trig functions and showing the equivalence of their geometric (unit circle) and ...

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