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Questions tagged [mg.metric-geometry]

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

4
votes
0answers
74 views

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
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
8
votes
0answers
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 ...
2
votes
1answer
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
1answer
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 $S_{r} := S \cap\B(p,r)$. ...
11
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3answers
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
0answers
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
2answers
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
1answer
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 ...
6
votes
1answer
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
1answer
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?
4
votes
0answers
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
1answer
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 ...
1
vote
0answers
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,...
4
votes
1answer
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 ...
3
votes
1answer
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
0answers
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
1answer
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
2answers
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
1answer
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
1answer
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 ...
6
votes
1answer
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
1answer
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 $Lip(r)\le L$. ...
3
votes
0answers
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 $...
7
votes
1answer
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 $\ell_\infty[0,1]$. ...
0
votes
1answer
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
2answers
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-...
1
vote
0answers
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 ...
5
votes
0answers
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 ...
0
votes
1answer
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 ...
4
votes
1answer
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 ...
1
vote
2answers
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 ...
3
votes
0answers
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 ...
5
votes
0answers
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
4answers
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
3answers
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 ...
8
votes
2answers
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 ...
13
votes
1answer
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
1answer
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 ...
5
votes
0answers
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:...
4
votes
2answers
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
0answers
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
1answer
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 ...
2
votes
0answers
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 ...
3
votes
1answer
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
8answers
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 ...
5
votes
0answers
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
1answer
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 $CAT(0)$ ...