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

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3
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0answers
55 views

Quasi-isometric quotients of finitely generated groups

Let $G$ be a finitely generated group, $S$ a finite symmetric system of generators of $G$, and let $H,K\subset G$ be subgroups. Then $S$ induces a metric on the quotients $G/H$ and $G/K$; for instance,...
11
votes
4answers
327 views

Tameness in $\mathbb{R}^{n^2}$ of the subset consisting of matrices of positive determinant

The Lie group $GL(n)$ being a manifold is locally path-connected. Consider its connected component of the identity $C\subseteq\mathbb{R}^{n^2}$. What is a good way of showing that $C$ is a tame ...
4
votes
1answer
147 views

Compact manifolds locally bi-Lipschitz to Euclidean space

I have a compact manifold $M$, and I am allowed to choose some Riemannian metric on it, exactly which I don't care. But I would love it if I could choose the metric $g$ such that every point has an ...
1
vote
0answers
37 views

What are the central points of a semi-nice region in the plane?

For a convex set in Euclidean space, there is an obvious notion of its center: namely, its center of mass, which by convexity lies in the set. For a nonconvex set there is just as obviously no nice ...
5
votes
1answer
247 views

When does there exist a convex polyhedron with given edge lengths?

Let $n$ be a positive integer, and let $n = \ell_1 + \dots + \ell_k$ be a partition of $n$. Then there exists a convex polygon with side lengths $\ell_1, \dots, \ell_k$ if and only if all of the $\...
5
votes
1answer
110 views

Are square tiled surfaces dense in the moduli space of translation surfaces?

I'm reading the survey "An introduction to Veech surfaces" by Pascal Hubert and Thomas Schmidt. At page 19 they state "In any fixed stratum, the set of square-tiled surfaces of that stratum is dense....
17
votes
3answers
475 views

Gromov-Hausdorff limits of 2-dimensional Riemannian surfaces

Let $\{M_i\}$ be a sequence of 2-dimensional orientable closed surfaces of genus $g$ with smooth Riemannian metrics with the Gauss curvature at least $-1$ and diameter at most $D$. By the Gromov ...
4
votes
2answers
132 views

Classification of symmetries of tilings in surfaces?

Is there a general study of the symmetries of tilings on surfaces? Conway, Goodman-Strauss & Burgiel classified them on $\mathbb S^2, \mathbb R^2$ and $\mathbb H^2$, with their 'Magic Theorem'. ...
4
votes
0answers
116 views

Classifying countable sets of weighted dots on a real line

Each dot is located on the real line and assigned a weight that can be positive or negative. A dot is equivalent to two(or more) dots located at the same place whose weights sum is equal to that of ...
2
votes
0answers
31 views

Is there a “last mile” criterion for a generalization of planar convex hulls to symmetric weighted graphs?

This question is motivated by an almost successful attempt to define planar convex hulls of a finite set of isolated points only via subset sums of the distances between point-pairs; the advantage of ...
10
votes
1answer
188 views

Open (resp., closed) balls homeomorphic to open (resp., closed) discs on the plane

Let $\Sigma$ be a compact (smooth) surface, with a geodesic metric $d$ (compatible with the topology of $\Sigma$). Let $x \in \Sigma$, and suppose you have the following: for every $r<1$, the open ...
2
votes
1answer
65 views

Characterizing left invariant and right-$O_n$ invariant distances on $GL_n$

Consider the group $GL_n(\mathbb{R})$ with its standard topology. It is not hard to show that there exists Riemannian metrics on it which are left-$GL_n$ and right-$O_n$ invariant. (In fact it's ...
1
vote
0answers
60 views

A third degree surface and a touching sphere [closed]

Let consider a surface $z=1/(xy)$ and a sphere defined by $(x-1.5)^2+(y-1.5)^2+(z-1.5)^2=3/4$. The sphere touches the surface at (1,1,1). Is it possible to prove that point (1,1,1) is the only ...
25
votes
2answers
1k views

Understanding sphere packing in higher dimensions

In a recent publication by the Ukrainian mathematician Maryna Viazovska the Kepler problem for dimension $8$ and $24$, namely the densest packing of spheres, was solved. Admittedly it is very ...
18
votes
1answer
754 views

Sphere packings : what next after the recent breakthrough of Viazovska (et al.)?

Given the march 2016 breakthrough concerning sphere packings by Viazovska for the case of dimension 8, and by Cohn, Kumar, Miller, Radchenko and Viazovska for the case of dimension 24, it follows that ...
4
votes
1answer
139 views

Distance comparison in submanifold versus in the underlying manifold

Let $(M,g)$ be the (underlying) manifold, $(S,g|)$ be a submanifold. Let $a,b,c \in S$. It's not in general true that $d_M(a,b)\leq d_M(a,c) \implies d_S(a,b)\leq d_S(a,c)$. QUESTION I: The above ...
2
votes
0answers
121 views

Throwing darts at a barn and putting a bullseye around them in higher dimensions

Let $X \in \mathbb R^d$ be a large domain (a ball of radius $r$ for $r$ large should suffice) Let $B$ be a ball of radius $1$. Consider the ratio $$ \frac{ \left| \left\{ x_1,\dots,x_n \in X \mid ...
1
vote
0answers
65 views

Finding the infimum using a piecewise isometry

Given a finite set of unit circles in the plane such that the area of their union $U$ is $S$, what is the largest possible bound $kS$ for some constant $k$ such that there exists a subset of mutually ...
14
votes
0answers
154 views

Precise estimate for probability an $n$-point set has diameter smaller than $1$

This question was inspired by an earlier question that I answered but would like a more precise bound for. Consider random points $x_1, \dots, x_n$ in the unit ball in $\mathbb R^d$, uniformly and ...
1
vote
0answers
166 views

A chain of six circles associated with six points on a circle (in Mobius plane) [closed]

I found a conjecture: A chain of six circles associated with six points on a circle (in Mobius plane). This is a generalization of the last my previous question in Three chains of six circles. (...
12
votes
2answers
257 views

Shortest path through $n^{1/3}$ points out of $n$

Say I sample $n$ points uniformly at random in the unit cube in $\mathbb{R}^3$, and then I look for the shortest path through $n^{1/3}$ of those points (rounding up, say). What happens to the length ...
3
votes
0answers
176 views

A conjecture on six planes [closed]

When I read Cox's Theorem, and Clifford's Circle Theorem and Miquel six circles theorem, I found the conjecture as folowing. And I checked the conjecture by the Geogebra sofware, the conjecture is ...
7
votes
1answer
480 views

A chain of six circles associated with a conic

I found this problems three years ago. But I never have been a proof. Recently I posted in math.stackexchange.com. I am looking for a solution of the following problems: A chain of six circles ...
16
votes
1answer
422 views

Just how close can two manifolds be in the Gromov-Hausdorff distance?

Suppose that we have two compact Riemannian manifolds $(M,g)$ and $(N,h)$. Define the Gromov-Hausdorff distance between them in your favorite way, I'll use the infimum of all $\epsilon$ such that ...
1
vote
0answers
39 views

Continuity of Busemann-Hausdorff area density

I am trying to find out why the Busemann-Hausdorff area density as defined by Burago and Ivanov is continuous. Here, $GC_m(V)\subset \Lambda^m(V)$ denotes the simple $m$-vectors in an $n$-dimensional ...
0
votes
0answers
64 views

Integral of gradient between level sets of Lipschitz functions

Start with a compact metric measure space $(X,d)$, with a doubling measure $\mu$ and a local regular Dirichlet form $\mathcal E$ that supports a Poincare inequality. $d$ can be taken to be the ...
10
votes
1answer
237 views

Surface area of an $\ell_p$ unit ball?

Are there any known formulas or approximations for the surface area of a unit ball in $d$ dimensions under the $\ell_p$ norm? As obvious examples, it is of course well-known that the surface area of ...
12
votes
0answers
107 views

Rational inscribed realization of the regular dodecahedron

While it is clear that the regular dodecahedron $D$ cannot be realized with all integer coordinates, it is easy to find a polytope, which is combinatorially equivalent (face lattice isomorphic) to $D$ ...
2
votes
1answer
208 views

Relation of some Euclidean geometry theorems and more conjecture generalizations

In this topic I want to share relation of the Pythagorean theorem, the Stewart theorem and the British Flag theorem, the Apollonius' theorem and the Feuerbach-Luchterhand. Since that I posed two ...
10
votes
0answers
782 views

Dao's theorem on six circumcenters associated with a cyclic hexagon

This questions from Ngo Quang Duong's paper In 2013, O. T. Dao published without proof a theorem with title Another seven circles theorem in Cut the Knot, a free site for popular expositionsof many ...
3
votes
1answer
129 views

Choosing $K$ “centers” from the space of permutations

Let $\Pi$ denote the space of all permutations of $\{1,\dots,n\}$, and let $d(\cdot,\cdot)$ be a metric on $\Pi$. Suppose I am given a large integer $K$ and I have to select $K$ permutations $\pi_1,\...
1
vote
1answer
63 views

Does a continuous function have a continuous integral function in a discrete dynamical system?

Let $X$ be a compact manifold (or the closure of a Euclidean domain if that helps significantly) and $T\colon X\to X$ a homeomorphism. Let us say that a function $v\colon X\to\mathbb R$ is the ...
4
votes
0answers
146 views

Questions on Thurston's metric on Teichmüller space

I'm reading the famous "Minimal stretch maps between hyperbolic surfaces" by William Thurston and I'm trying to understand the key theorem 8.1. I have many unclear points so I hope someone can help me ...
1
vote
1answer
151 views

Applying Cheeger and Colding segment inequality

The question turns out quite long and maybe a bit vague, I apologize in advance for that. I am currently trying to understand Cheeger and Colding proof of the almost splitting theorem. Currently I ...
8
votes
1answer
150 views

Convex body with affine-equivalent cross-sections

I recently discovered the following fact: Let $K\subset\mathbb R^3$ be an origin-symmetric convex body with smooth and strictly convex boundary. Suppose that all central cross-sections of $K$ (that is,...
1
vote
0answers
62 views

Possible directions of saddle connections

Let's consider a Riemann surface $X$ of genus $g\ge 2$ and $q$ a holomorphic quadratic differential on $X$. A natural parameter on $X$ is a chart for which $q=dz^2$. A $\theta$-trajectory is a maximal ...
8
votes
3answers
271 views

Shape whose translated and scaled copies are closed under intersection

The translated and scaled copies of an equilateral triangle with fixed orientation are closed under intersection - the intersection is again an equilateral triangle with the same orientation. What ...
3
votes
0answers
64 views

Name for metric spaces with useful unique-ball-intersection property?

When dealing with the problem of extending a Lipschitz function $f:A \to Y$ between metric spaces across an inclusion $A \hookrightarrow X$, one often imposes (conditions which imply) the following ...
4
votes
1answer
265 views

Sixteen points circle - A conjecture on Möbius plane

The conjecture refer the reader about the Bundle's theorem configuration. (This conjecture from a note) Consider the Bundle theorem configuration : Points $A_1, A_2, A_3, A_4$ lie on a circle, ...
7
votes
0answers
126 views

Isometry group of low dimensional Alexandrov space

It is known by the work of Galaz-García and Guijarro, that the dimension of the isometry group of an $n$-dimensional Alexandrov space (of curvature bounded below) is bounded above by $\frac{n(n+1)}{2}$...
4
votes
2answers
270 views

k nearest points

Assume $n$ points $P_i \in \mathbb{R}^2, i \in {1,2,...,n}$. For each point there is a $k$ nearest neighbour $(k<n)$, or equivalently for each point $P_i$ there is one circle with center the point $...
3
votes
1answer
138 views

Moment matching on the standard simplex

Let $\vec{\mu}_1, \vec{\mu}_2,\ldots, \vec{\mu}_k \in \Delta^{d-1}$ be $k\ (k\geq 2)$ distinct vectors on the standard simplex, where $$\Delta^{d-1} = \{\vec{\mu}\in R^{d}:\| \vec{\mu}\|_1 = 1,\mu_j \...
12
votes
2answers
261 views

Term for a metric space for which the triangle inequality is strict?

Is there a standard term for a metric space in which $\rho(p,r) < \rho(p,q) + \rho(q,r)$ for any distinct $p$, $q$, $r$? Sort of the opposite of metric convexity. For instance, a subset of ...
7
votes
1answer
202 views

Partitioning a convex object without harming existing convex subsets

$C$ is a convex planar figure and $C_1,\dots,C_n$ are pairwise-disjoint convex subsets of $C$, like this: A convex-preserving partition of $C$ is a partition $C=E_1\cup\dots\cup E_N$, , such that $...
10
votes
0answers
157 views

Electrons on a pancake ellipsoid

The problems of minimizing the potential energy of electrons on a sphere, or maximizing the smallest distance between the electrons, have been well-studied. E.g., see the earlier MO question "...
12
votes
1answer
247 views

Covering the unit sphere by open hemispheres

Suppose $H_1,\ldots,H_{2n}$ are open hemispheres which cover $S^{n-1}$ with the property that removing any one of them leaves $S^{n-1}$ uncovered. Is it necessarily the case that the hemispheres can ...
1
vote
0answers
60 views

Comparison theorem for Lambert quadrilateral

A Lambert quadrilateral is a quadrilateral three of whose angles are right angles. And in 2-d hyperbolic space $\mathbb H^2$, we have nice formulas for the fourth angle. If $AOBF$ is a Lambert ...
0
votes
0answers
37 views

Minimize a function to learn a mapping

I have two questions. I want to learn a mapping $M$ that minimizes the right-hand side of the following equation: $E =\frac{1}{N} \sum\limits_{i=1}^N \bigg(\sum\limits_{j=1}^K \alpha_i - \big(\...
5
votes
0answers
178 views

Hausdorff dimension of Apollonian circle packing, 1.305686729, 1.305688 or yet something else?

I am interested in the Hausdorff dimension of the Apollonian circle packing. There seem to be two numerical calculations of the value: 1.305686729(10) from P.B ...
-1
votes
1answer
115 views

Construction of fibration over Riemannian Manifold

Let $\pi: E \rightarrow B$ be a fibration over a Riemannian manifold $B$, with $\pi^{-1}[b]$ homeomorphic to $\mathbb{R}$. More precisely: I want each fiber $\pi^{-1}[b]=Im(f_b)$ for some $C^{\...