# Questions tagged [mg.metric-geometry]

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

3,758
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### What will be the upper bound of covering number of $r B_{n}^2$?

I am very curious to know the covering numbers of the $r$-radius Euclidean ball $B_{n}^2$ for any $r > 0$: so
$r B_{n}^2$ is a Euclidean ball with radius $r$.
I have read Covering number of $l_2$ ...

1
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52
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### Measure estimates of $\delta$-neighbourhood of compact sets

I am interested in the estimating from above the measure of a compact set $K$ by a sequence of sets $K_n$, converging to it in the Hausdorff metric. As such I am looking for known conditions that give ...

10
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436
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### Does a compact contractible metric space have a point that is fixed by all isometries?

Let $(X,d)$ be a compact and contractible metric space. Let $\operatorname{Isom}(X)=\{\phi\colon X\to X\}$ be its group of isometries.
Question: Is there a point $x\in X$ fixed by all $\phi\in\...

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### Symmetries of contractable subsets of $\Bbb R^n$

Let $K\subset\Bbb R^n$ be a non-empty compact subset of $\Bbb R^n$. A symmetry of $K$ is an isometry of $\Bbb R^n$ that fixes $K$ set-wise. Since $K$ is compact, there is always a point $x\in\Bbb R^n$ ...

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63
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### Closed almost geodesics in a Riemannian manifold

Let $M$ be a smooth Riemanniann manifold. For $\varepsilon \geq 0$ we call an $\varepsilon$-geodesic (I am not sure that this is a standard name) a smooth map
$$\gamma\colon [a,b]\to M$$
such that for ...

1
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1
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53
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### Lower bound on a norm of $\mathbb{CP}^2$ inducing a lower bound on the Euclidean norm of $\mathbb{C}^3$

Let $|\cdot|$ denote the usual Euclidean norm on $\mathbb{C}^3$ and fix some arbitrary metric $\rho$ on $\mathbb{CP}^2$. For $\delta > 0$ and any set $\hat{P} \subset \mathbb{CP}^2$, define the $\...

12
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+50

### Can you perturb an inscribed polytope so all its edges grow?

Consider the family of convex simplicial polytopes with vertices in the unit sphere of $\mathbb{R}^n$ which have the origin as an interior point.
My question is the following:
Let $P, P'$ be two non-...

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38
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### Smallest doubling subset of a set in a metric space

Let $(X,d)$ be a separable metric space and $A\subseteq X$ be compact.
Since every finite set is doubling then, the collection $\mathcal{A}$ of doubling subsets of $A$ cannot be empty. My initial ...

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87
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### A neighborhood $Y$ of a set $X$ such that the line segment connecting any point in $Y$ and its projection to $X$ is contained in $Y$

A direct line from a point $p$ to a set $X$ is a line segment with one endpoint at $p$ and one endpoint in $X$, which is as short as any other line segment from $p$ to $X$. Given a closed set $X$ and ...

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

Define the Jaccard distance between two continuous vectors $a, b\in [0,1]^p$ as
\begin{equation}
J(a,b) = 1 - \frac{\|a\odot b\|_1}{\|a\odot b\|_1+\|a-b\|_1}
\end{equation}
where $\odot$ is the ...

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### Cylinder orientation representation

I'm trying to find an efficient computation and representation for the following problem.
Given a cylinder with height $h$ and radius $r$ with a given position $\mathbf{x} = [x, y, z]$ and $N$ number ...

2
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1
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93
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### Concyclic point made from Six arbitrary points

Let $A_1A_2A_3A_4A_5$ be irregular convex Pentagon and Let $P$ be arbitrary point anywhere in Plane geometry.
Let $X_1,X_2,X_3,X_4,X_5$ be Circumcircle of $\triangle PA1A3$; $\triangle PA2A4$; $\...

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202
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### Questions about John Lott's paper in 2000

I'm reading Lemma 4.1 of Lott, John, A hat-genus and collapsing., J. Geom. Anal. 10, No. 3, 529-543 (2000). ZBL1047.53024. I'm a little confused with the Atiyah-Patodi-Singer boundary condition. Let $...

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86
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### Question about deformation of the metirc on a Riemannian manifold

I'm a bit confused with the deformation of the metric on a given Riemannian manifold $(M,g)$ with a smooth boundary. How can we deform the metric $g$ such that it is a product near $\partial M$, ...

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93
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### $O(n^{2-\epsilon})$ bound on choosing $n$ points on the hypersphere to maximize $\pm 1$ weighted sum of their $\binom{n}{2}$ inner products

Given $n,d\in\mathbb{N}, n\gg d$, I'm looking for a bound on the maximum (or minimum) expected value of the following game:
Draw a vector $\epsilon\in\{\pm 1\}^{\binom{n}{2}}$, uniformly at random. ...

1
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1
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66
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### When is the mode of a stochastic process a better statistic than the mean?

This is a soft question.
I've been interested in Onsager-Machlup theory recently. Essentially, the Onsager-Machlup function serves the role of a density but it can exist on non locally compact spaces.
...

3
votes

1
answer

119
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### Sphere in Urysohn space

Let $S$ be a unit sphere in the Urysohn space $U$.
Is it true that any isometry $S\to S$ can be extended to an isometry $U\to U$?

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1
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38
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### Convergence of extremal subsets in Alexandrov spaces

Let $\{X_i^n\}$ be a sequence of $n$-dimensional Alexandrov spaces with curvature uniformly bounded from below which converges in the Gromov-Hausdorff sense to a compact $n$-dimensional Alexandrov ...

32
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16
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### Which theorems have Pythagoras' Theorem as a special case?

Loomis famously wrote hundreds of proofs of Pythagoras' Theorem (reference below), but these are all basically proofs "from below". Today on Twitter @panlepan mentioned Carnot's theorem ...

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128
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### Projecting a given point onto a random $2$-dimensional plane in more than $3$ dimensions

We are given $\mathbf{p}\in\mathbb{R}^d$, where $d\gg 1$. Let $\mathbf{v}$ be a point selected uniformly at random from the unit $(d-1)$-sphere $\mathcal{S}^{d-1}$ centered at the origin $\mathbf{0}\...

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### Connection between a function and its usage in geometry [closed]

I know nothing about geometry, but I found a function which seems to have something to do with geometry.
This function is, $$f(x,y,z) = \dfrac{(x,y,z)}{\sqrt{1 + x^2 + y^2 + z^2}}$$
where $x,y,z$ is ...

3
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1
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108
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### Convex sphere in R^3

Is every convex sphere (in the sense of Alexandorff, which is the boundary of some convex body in $\mathbb{R}^3$) with Alexandorff curvature $\geq 1$, bi-Lipschitz to the unit round sphere in $\mathbb{...

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### Roundest polyhedra: how well can we bound the edge count of their faces?

By "roundest" I mean having the lowest surface area for the highest volume, given a fixed number of faces $n$. There've been a few questions about them on here (including from me), but I'm ...

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145
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### What must a set of $n$ points in 2D space fulfill so that it is possible to connect them through tangent circles

In high-school I learned how to find the circles that connect points in 2D space forming a curve made out of tangent circles like this:
(The green line shows the "initial direction" of the ...

4
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93
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### Ends of a negatively curved Riemannian manifold

Let $M$ be a complete Riemannian manifold. Let us use the standard definition of "end", for example, as in this article. If $M$ has non-negative Ricci curvature, it is well-known that it has ...

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103
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### What is a closed monoidal metric space?

This recent MSE question started a conversation with the OP of that post about what are some categorical notions casted in the category of metric spaces, regarded as enriched categories over $[0,\...

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### Fixed points on spherical buildings

A crucial aspect of the Bruhat–Tits theory of affine buildings is the Bruhat–Tits fixed-point theorem, which, in one of many formulations, states that, if $\Gamma$ is a group of isometries of an ...

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352
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### Functions $\mathbb{R}^2\to\mathbb{R}^2$ that preserve lines

The simplest case of the Fundamental Theorem of Projective Geometry states that, if $f: \mathbb{R}^2\to\mathbb{R}^2$ is a bijection that preserves lines – in the sense that if $L\subseteq\mathbb{R}^2$ ...

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### Are Carnot groups ever CAT(𝜅) spaces?

Let $G$ be a free Carnot group of homogeneous dimension $d$, equipped with the Carnot–Carathéodory metric. Is $(G,d)$ ever $\operatorname{CAT}(\kappa)$ for some $\kappa\in \mathbb{R}$?

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### Open covering with bounded diameters [closed]

Here is an interesting puzzle I came across.
I have no idea which tools could be applied to solve it, so the tags may be misleading.
For any $A \subseteq \mathbb{R^n}$ , its diameter is defined by
$$\...

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1
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218
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### Can every smooth space curve be realized as an origami curved crease?

Many years ago, Ron Resch told me that he proved that every smooth simple space curve $C$ could be realized as a curved crease $\gamma$ in the interior of a piece of paper.
He never published this (as ...

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145
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### Do all manifolds admit metrics with Euclidean balls?

Let $M$ be a compact topological n-manifold. Suppose we are given a locally flat embedding $M \subset \mathbb{R}^{n+k}$. This induces a metric on $M$ by restriction. Is it true that for $\epsilon$ ...

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### Can every non-hemi uniform polytope tile hyperbolic space?

Can every uniform polytope which is not a hemipolytope tile hyperbolic space on its own?

5
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208
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### Manifolds with nonpositive radial curvature

How can one construct examples of Riemannian manifolds which have nonpositive radial curvature about some point, but are not nonpositively curved everywhere? (I presume that they exist, but do not ...

2
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### Convergence in Hausdorff distance of intersection of closed linear subspaces with a given closed convex set

I've run into the following problem when doing some work with non-commutative metric spaces, which seems like something people may have thought about before but I can't find anything on this problem ...

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54
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### Maximizing a parametric integral over the unit sphere

I am trying to compute the nonnegative quantity
$$
\underset{y\in\mathbb{S}^{d-1}}{\sup}\int_{0}^{t}(\Vert A(\tau)y\Vert_{1}- \Vert A(\tau)y\Vert_{q})d\tau, \quad 1 < q < \infty
$$
where $\...

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0
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69
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### About convex functions

Is there a non negative, convex, and decreasing function $g$ on $[0,\infty)$, with $g(0)=1$, such that $g(s+t)< g(t)g(s)$ for $s,t \in (0,\infty)$?

1
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1
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173
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### Probability that three vectors of a unit sphere lie on one side of a hyperplane if angle between the vectors are given

As the title says, How to find the probability of vectors a, b, c, on some unit sphere, all lies on same side of some hyperplane passing through the origin. Information present are the angles between ...

3
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### (Homotopy) colimit and manifold

Suppose that I have an arbitrary regular CW complex. By associating a topological space to each vertex of the CW complex, I can have a diagram of topological spaces, denoted by $D$, over the CW ...

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### Weak compactness in $\mathcal{F}(X)$

Let $(X,0)$ be a pointed metric space and let $\mathcal{F}(X)$ be the natural predual of ${\rm Lip}_0(X)$, the space of Lipschitz functions on $X$ that map $0$ to $0$; here $\mathcal{F}(X)$ is really ...

23
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### Polyomino that can cover an arbitrarily large square but not the entire plane

https://userpages.monmouth.com/~colonel/nrectcover/index.html
For a polyomino with no holes that cannot tile the plane, we may ask what are the maximal rectangles and infinite strips that it can ...

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### Any round convex body between a simplex and a ball?

Fritz John's ellipsoid theorem gives the minimal ball containing a given convex body $K$. Moreover, we have for $m$ points in boundary
$$\tag{$A$}\label{A}\sum_m c_iu_iu_i^{T} = I_n,$$
where $I_n$ is ...

7
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146
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### Tiling space with supertile of hypercube unfoldings

Two students in my class
asked and answered what might be a novel question.
It is well known that the cube has exactly $11$ edge-unfoldings
(or "nets"), as shown below:
(Image from ...

5
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62
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### Concentration bound on additive functions with constraints

Given a family of sets $F \subseteq P(\{1,\ldots,n\})$. I define the function $f_F:[0,1]^n \rightarrow R$ to be $f_F(x_1,\ldots,x_n)= \max_{S \in F} \sum_{j \in S} x_j$.
Given a series of independent ...

3
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1
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98
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### Talagrand's inequality for L1 norm

I have a series of $n$ independent random variables $X_1,\ldots, X_n$, each with the support $[0,1]$, and a monotone convex function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ that is 1-Lipshitz in L1 ...

1
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0
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33
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### Are cells of 4-polytopes a convex polyhedron by definition?

I'm going by the Wikipedia definition for a 4-polytope.
Do by definition, cells of 4-polytopes have to be a convex polyhedra?
If not, then are there polyhedra with non-convex faces?
If yes, is it the ...

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0
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106
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### Relation between the distance projective maps and their angles

Let $f:N \to \mathbb{R}^2$
be a differentiable map of smooth manifolds. Let $\mathbb{R}^2$ be decomposed as a direct sum of line bundles, i.e. $\mathbb{R}^2=E(x) \oplus F(x)$, where $F(x)$ and $E(x)$ ...

7
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227
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### Spectral radius of a finitely generated group

Let $G$ be a finitely generated group and $\Gamma$ be its Cayley graph with the usual word metric. Let $\mu$ be a symmetric non-degenerate measure on $G$ (maybe with finite support or smooth), and ...

7
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1
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203
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### Finding maximal prefix of a simple curve

Let $S$ be a piecewise-linear simple curve. For a point $p_1$ on $S$ I want to determine a maximal interval $I$ on $S$ starting in $p_1$ such that $I$ is contained in a unit disk. Is this possible, or ...

0
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43
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### Examples of metric entropy of convex bodies in $\mathbb{R}^n$

I am interested in examples of convex bodies in $\mathbb{R}^n$ whose metric entropy in terms of the Euclidean norm has been characterized.
Specifically if $N(K, \|\|_2, \varepsilon)$ denotes the ...