# Questions tagged [mg.metric-geometry]

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

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### Computing the volume of intersection between a hyper-rectangle and a ball

$C$ is a region bounded above, below coordinate-wise by $\overline{c},\ \underline{c}\in [-1,1]^d.$ Is there a procedure not exponentially complex with $d$ that computes the volume of $C\cap B(0,1)$ ...

**3**

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**0**answers

98 views

### Bound on change in relative length from 'well-behaved' Jacobian?

(This question was originally asked on Mathematics Stack Exchange, and sat there for several weeks with low views and no answers.)
Let $\phi$ and $\gamma$ be rectifiable curves in the same length ...

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**0**answers

42 views

### n-D Gauss circle problem over a rectangle

I would like to approximate the amount of points in $\left(2^{-a\cdot n}\mathbb{Z}^n\right)\cap B^n(0,1)\cap C^n$ where $a>0,\ B^n(0,1)$ is the unit nball and $C^n$ is some rectangular domain ...

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votes

**0**answers

34 views

### Is volume of abstract polytope realisation bounded by length of edges?

Suppose we have abstract polytope F of dimension d ( that is the greatest rank facet has rank n). Such abstract object may have realisations in d-dimensional Euclidean space as polytopes $A_i(F)$, and ...

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**0**answers

130 views

### Geodesics between boundary points of a hyperbolic space

Let $X$ be a (not necessarily proper) hyperbolic space. Following Gromov, we define the boundary of $X$ as the set of equivalence classes of sequences convergent at infinity. In general, it is not ...

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votes

**1**answer

50 views

### Convexity of set of normal directions in a CAT(0)-space

Let $X$ be a $\mathrm{CAT}(0)$ space, $p\in X$ and $v\in T_pX$. Let $N\subset T_pX$ be the set of tagent vectors making an angle greater than or equal to $\pi/2$ with $v$.
Is it true that the set $\...

**3**

votes

**1**answer

116 views

### Relation between a distance function and normal coordinations

$
\newcommand{dist}{\operatorname{dist}}
\newcommand{B}{\mathbb{B}}
$
Let $\mathcal {M}$ be a Riemannian manifold, $p \in S \subset \mathcal{M}$ and $r>0$. Denote $S_{r} := S \cap\B(p,r)$.
...

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votes

**3**answers

2k views

### Are uniformly continuous functions dense in all continuous functions?

Suppose that $X$ is a metric space. Is the family of all real-valued uniformly continuous functions on $X$ dense in the space of all continuous functions with respect to the topology of uniform ...

**5**

votes

**0**answers

235 views

### Almost monochromatic point sets

There are many sort of equivalent theorems about monochromatic configurations in finite colorings, such as Van der Waerden, Hales-Jewett or Gallai's theorem, the latter of which states that in a ...

**5**

votes

**2**answers

242 views

### An integral involving three Bessel functions

I am looking for a closed form for the following integral
$$ I = \int_0^\infty \mathrm{d} x \ x \ J_0(ax) J_0(bx) J_1(cx) $$
which can be thought of as a particular case of the more general integral
...

**7**

votes

**1**answer

348 views

### Homeomorphism/ homotopy types of non-negatively curved manifolds

A (special case of a) theorem of Gromov says for any $n\in \mathbb{N}$ there exists a constant $C(n)$ such that for any smooth connected closed $n$-dimensional Riemannian manifold with non-negative ...

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votes

**1**answer

171 views

### Triangulations of convex surfaces

Let $M$ be a smooth closed positively curved surface in Euclidean 3-space, $T$ be a geodesic triangulation of $M$, and $E$ be the edge graph of the convex hull of vertices of $T$.
It is easy to see ...

**5**

votes

**1**answer

124 views

### Fundamental group of Alexandrov space.

Is it true that the fundamental group of a compact finite dimensional Alexandrov space with curvature bounded below is finitely generated?

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**0**answers

97 views

### “Uniqueness” of the Yamaguchi submersions of Riemannian manifolds

The Yamaguchi submersion theorem says the following. Let $\{M_i\}$ be a sequence of $n$-dimensional smooth connected closed Riemannian manifolds of diameter at most $D$ and sectional curvature at ...

**6**

votes

**1**answer

317 views

### Bichromatic pencils

A pencil is a collection of some lines through a point, called the center of the pencil.
If the points of the plane are colored, then call a pencil bichromatic if there is a color that is present on ...

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vote

**0**answers

65 views

### Integral of the square of the areas of slices of a shape

Suppose $\omega$ is a bounded shape in $\Bbb{R}^3$ and that $\{z : (x,y,z) \in \omega \}=[0,T]$ (that is, the shape is exactly contained in the band $\{z \in [0,T]\}$. If we denote by $\omega_t = \{(x,...

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**1**answer

267 views

### Canonical Metrics on 3- and 4-Manifolds

From the Uniformization Theorem, it is known that every conformal class of metrics on a genus-$g$ Riemann surface with $n$ boundaries/punctures, subject to the condition $2g+n\ge 3$, contains a unique ...

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votes

**1**answer

83 views

### Relative to Isoperimetric inequality with n-polygon

Since Brahmagupta's formula and Bretschneider's formula we have the inequality:
Any two quardrilaterals $A_1A_2A_3A_4$ and $B_1B_2B_3B_4$ with the same sidelengths and $A_1A_2A_3A_4$ is a cyclic ...

**5**

votes

**0**answers

126 views

### Filling points to a simplex in models for EG

I have a question which is related to higher Dehn functions of groups.
I also have a group $G$ with a finite $K(G,1)$. Let us denote by $EG$ the universal cover of this complex. We choose a path-...

**9**

votes

**1**answer

384 views

### Strengthened version of Isoperimetric inequality with n-polygon

Let $ABCD$ be a convex quadrilateral with the lengths $a, b, c, d$ and the area $S$. The main result in our paper equivalent to:
\begin{equation} a^2+b^2+c^2+d^2 \ge 4S + \frac{\sqrt{3}-1}{\sqrt{3}}\...

**10**

votes

**2**answers

312 views

### Tangled random triangles: One giant component?

Suppose you have $n$ triangles whose corners are random points on a sphere $S$
in $\mathbb{R}^3$.
Viewing the triangles as built from rigid bars as edges,
two triangles are linked if they cannot be ...

**10**

votes

**1**answer

288 views

### Odds on rolling a rhombicosidodecahedron

This is more of a curiosity to me, but I'm sure I don't have the mathematical skills to answer it. That said... I took a look at several other posts with questions that relate to this one, but I ...

**1**

vote

**1**answer

118 views

### Wasserstein distance to the set of Gaussians ; Boltzman dissipation rate

I am interested in the $2$-Wasserstein distance for probabilities over ${\mathbb R}^n$,
$$W_2(\mu,\nu)=\left(\inf\int_{{\mathbb R}^n\times{\mathbb R}^n}|w-v|^2\pi(v,w)\right)^{1/2}$$
where the infimum ...

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votes

**1**answer

184 views

### Embedding the $\ 2^n+1$-point metric 1-space

A metric space $\ (S\ d)\ $ is said to be a 1-space $\ \Leftarrow:\Rightarrow\ \forall_{x\ y\in S}\ (x\ne y\ \Rightarrow\ d(x\ y)=1).$
Question: Do there exist a non-negative integer $n,\ $ ...

**5**

votes

**1**answer

226 views

### Is each compact metric space a subset of a compact absolute 1-Lipschitz retract?

A metric space $X$ is called an absolute $L$-Lipschitz retract if for any metric space $Y$ containing $X$ there exists a Lipschitz retraction $r:Y\to X$ with Lipschitz constant $Lip(r)\le L$.
...

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votes

**0**answers

87 views

### Is the Banach space $C(K)$ a $1$-Lipschitz comp-extensor?

Given a real number $c\ge 1$ let us say that a metric space $X$ is a $c$-Lipschitz comp-extensor if each Lipschitz self-map $f:K\to K$ of a compact subset $K\subset X$ extends to a Lipschitz self-map $...

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votes

**1**answer

151 views

### What is the smallest Lipschitz constant of a Lipschitz retraction of $\ell_\infty([0,1])$ onto $C[0,1]$?

By Theorem 1.6 in the book "Geometric Nonlinear Functional Analysis" by Benyamini and Lindenstrauss, the Banach space $C[0,1]$ is a Lipschitz retract of the Banach space $\ell_\infty[0,1]$. ...

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votes

**1**answer

78 views

### The radius of an interval mapped through a space-filling path

Take $f:[0,1]\to [0,1]^n$ a continuous tour through $[0,1]^n,$ say, some iteration of a Hilbert curve. For $\varepsilon \in (0,1)$ what is the following thing called and are there any nontrivial upper ...

**39**

votes

**2**answers

3k views

### Can one “hear” the shape of a polygon via external reflections?

This question is a rough analog of Kac's "Can One Hear the Shape of a Drum?"
A closer analog is the recent "Bounce Theorem" that says, roughly, the shape of a polygon is determined by its billiard-...

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vote

**0**answers

86 views

### Is it possible to dissect a regular polygon into mirrored-symmetric pieces?

Q1. Planar regular triangle is dissected into three congruent pieces, each of them having no symmetry axis.
Can it be so, that one of these pieces is a mirrored (and then rotated) copy of some other ...

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votes

**0**answers

142 views

### Question related to high dimensional kissing number

I have a question related to the kissing number in $n$ dimension.
Suppose we have many non-overlapping $n$-dimensional balls of radius $1/2$. We place one of the $1/2$-radius ball centered at the ...

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votes

**1**answer

303 views

### Metric space that is not a subspace of $\mathbb{R}^n$? [closed]

What are some simple examples of metric spaces that cannot be subspaces of $\mathbb{R}^n$? I've heard there is an example with $4$ points, where two points lie between the other two, but I cannot ...

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votes

**1**answer

387 views

### Arranging squares without overlap

What is the smallest positive real $r\in\mathbb{R}$ with the following property?
Every finite collection of squares such that the sum of their areas equals $1$ can be arranged without overlap ...

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vote

**2**answers

119 views

### For what points in $\mathbb R^2$ does the triangular condition fail for a 1/2-metric?

It is known that for $p<1$, $d_p(x,y)=\big(\sum_{i=1}^n |y_i-x_i|^p\big)^{1/p}$ is not a metric. In the case of 2 dimensions and $p=1/2$ it seems rather hard to find a counterexample where the ...

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**0**answers

115 views

### Does every non-locally compact metric space admit a violation of Lebesgue's theorem?

From the results of Preiss and Tišer, it is known that many natural families of measures on Hilbert spaces violate the Lebesgue Density Theorem. Question: Does every non-locally compact metric space ...

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**0**answers

203 views

### Longest simple path through hypercube corners

This is a variation on a previously answered question,
Longest path through hypercube corners.
Here I am seeking the longest simple (non-self-intersecting) path through
the unit hypercube's vertices,
...

**12**

votes

**4**answers

732 views

### Longest path through hypercube corners

Is the longest Hamiltonian path through the $2^d$ unit hypercube vertices known,
where path length is measured by Euclidean distance in $\mathbb{R}^d$?
The unit hypercube spans from $(0,0,\ldots,0)$ ...

**9**

votes

**3**answers

495 views

### Non embedding of the Heisenberg group

It is well known that Heisenberg groups cannot be bi-Lipschitz embedded into Euclidean spaces. A standard proof uses the fact that a Lipschitz mapping from a Heisenberg group into a Euclidean space is ...

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votes

**2**answers

269 views

### A characterization of metric spaces admitting a bi-Lipschitz embedding into a Hilbert space?

Theorem (??) derived in this MO-post from Schoenberg's theorem yeilds a "bipartite" characterization of metric spaces that admit an isometric embedding into a Hilbert space. This Theorem (??) implies ...

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votes

**1**answer

256 views

### A reference to a characterization of metric spaces admitting an isometric embedding into a Hilbert space

I am looking for a reference to the bipartite version of the Schoenberg's criterion of embeddability into a Hilbert space. The Schoenberg criterion is formulated as Proposition 8.5(ii) of the book "...

**3**

votes

**1**answer

102 views

### Question arise from kissing number in 2 dimension

I'm considering an extended problem of kissing number in $\mathbb{R}^2$.
Suppose I have a given disc $\mathcal{D}$ of radius 1/2 and infinitely many discs all of radius 1/2 and all these discs and ...

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votes

**0**answers

261 views

### When are Lipschitz functions dense in continuous functions?

Let $X$ be a compact metric space, and let $Y$ be another metric space.
I am looking for examples of, and especially references to, theorems that give conditions under which any continuous mapping $f:...

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votes

**2**answers

193 views

### Doubling dimension vs other metric dimensions

For separable metric spaces, three fundamental notions of dimension
are equivalent:
$$ \text{dim }X = \text{Ind }X = \text{ind }X ,$$
Where does the doubling dimension
fit into the picture?

**6**

votes

**0**answers

128 views

### Sets of points avoiding small angles

(1) $\mathbb{R}^2$.
I'd like to place $n$ points in the plane so that the smallest angle they
determine is as large as possible.
In a sense, such a point set is in very general position, not only
...

**1**

vote

**1**answer

123 views

### How does Siegel's Hilbert-Blumenthal fundamental domain differ from Götsky's?

This question has been changed to something related but different from the original question. Thanks to @paulgarrett for chatting with me and helping me hone in on a more interesting part.
The first ...

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**0**answers

66 views

### Is there any way to deform a Riemannian metric while keeping a positive curvature lower bound?

This is a somehow general question. Suppose you have a Riemannian manifold with sectional curvature lower bound, say $\kappa$. We want to find a way to deform the metric in a way such that the lower ...

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votes

**1**answer

115 views

### Smooth approximation of locally CAT(-1) metrics

It is a well known fact that locally CAT(-1) metrics on surfaces can be approximated by hyperbolic polyhedral metrics with cone singularities: roughly speaking you pick a geodesic triangulation of the ...

**20**

votes

**8**answers

3k views

### Mathematical theory of aesthetics

The notion of beauty has historically led many mathematicians to fruitful work. Yet, I have yet to find a mathematical text which has attempted to elucidate what exactly makes certain geometric ...

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**0**answers

148 views

### Local isometry of complete length spaces that is not a covering map

Let $\pi:\widetilde{M}\to M$ be a surjective local isometry between complete length spaces (local isometry means that every point $x\in \widetilde{M}$ has a neighborhood which is isometrically mapped ...

**6**

votes

**1**answer

205 views

### Bruhat-Tits building of $SL_n(\mathbb{Q})$, hyperbolic isometries and its axis

Consider $G=SL_n(\mathbb{Q})$ and $p$ a prime integer. Associated to $G$ and $p$ we have its Bruhat-Tits building $\Delta$.
It is well known that $\Delta$ can be provided with a canonical $CAT(0)$ ...