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

learn more… | top users | synonyms (1)

2
votes
1answer
80 views

Coordinate chart of concave functions near a regular point in Alexandrov spaces

Let $M$ be an Alexandrov space with curvature $\geqslant -1$. Then we have the following theorem which is often used to perturb a regular point to points we want. Let $g_0$ be a ...
6
votes
1answer
211 views

Polygons in the Cartesian plane

Consider the octagon in the Cartesian plane with vertices at (0,0), (1,0), (1,2), (4,2), (4,6), (7,2), (7,8), and (0,8). Are there other (infinitely many) polygons, such as this, lying entirely in ...
8
votes
2answers
900 views

Wasserstein distance in R^d from one dimensional marginals

This question occurred to me while I was reading Klartag's papers on central limit theorems for convex bodies. Given probability measures $\mu$, $\nu$ on (the Borel $\sigma$-field of) $R^d$ with ...
0
votes
1answer
71 views

The pointwise Lipschitz-ness of a function on a dense set, implies its pointwise Lipschitz-ness everywhere?

Some definitions: Let $(M,d)$, $(M',d')$ be metric spaces. For $f:M\to M'$, $x\in M$ and $r>0$, define $$D_r(f)(x):= \sup\{r^{-1}d'(f(x),f(y)): y\in M,\,d(x,y)\leq r\}.$$ Define the pointwise ...
17
votes
3answers
413 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 ...
2
votes
1answer
77 views

Volume of the subelliptic ball

Let $\Omega \in \mathbb{R}^n$ a bounded open set when $n\geq 2$, and let $X_{1},X_{2},\cdots,X_{m}$ be real smooth vector fields that satisfy Hormander condition on $\Omega$. If we denote $Q(x)$ as ...
4
votes
0answers
64 views

Rational $d$-simplices

Define a rational $d$-simplex as a simplex in $\mathbb{R}^d$ such that the measure of all its $k$-dimensional faces, $k \ge 1$, is rational. So a rational triangle has rational edge lengths and ...
1
vote
0answers
44 views

A centralised website for computational attemps in graph theory and metric geometry?

The set of questions below stems from this question. 1) does a website exist that contains (at least links to) code and data files, with the aim to centralise computational results in graph ...
3
votes
1answer
133 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 ...
3
votes
0answers
52 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 ...
11
votes
4answers
310 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 ...
3
votes
5answers
2k views

Distance of a barycentric coordinate from a triangle vertex

I have a triangle $ABC$ with side lengths $a,b,c$ (edges $BC, CA, AB$ respectively). I have a point $p$ with barycentric coordinates $u:v:w$. These are normalised: $u+v+w=1$. $1:0:0$ corresponds to ...
36
votes
2answers
2k views

Shortest closed curve to inspect a sphere

Let $S$ be a sphere in $\mathbb{R}^3$. Let $C$ be a closed curve in $\mathbb{R}^3$ disjoint from and exterior to $S$ which has the property that every point $x$ on $S$ is visible to some point $y$ of ...
1
vote
0answers
28 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
242 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 ...
4
votes
1answer
129 views

Does Alexandrov space satisfy a reverse doubling condition?

Let $X$ be an $n-$dim Alexandrov space with curvature $\geq k$. Does $X$ satisfy a reverse doubling condition? That is, does there exist a constant $C>1$, s.t., for any $x\in X$, ...
3
votes
1answer
93 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 ...
5
votes
1answer
98 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 ...
4
votes
2answers
119 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'. ...
64
votes
4answers
5k views

Is the sphere the only surface all of whose projections are circles? Or: Can we deduce a spherical Earth by observing that its shadows on the Moon are always circular?

Several ancient arguments suggest a curved Earth, such as the observation that ships disappear mast-last over the horizon, and Eratosthenes' surprisingly accurate calculation of the size of the Earth ...
4
votes
0answers
114 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
1answer
62 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 ...
5
votes
0answers
476 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 ...
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 ...
15
votes
1answer
518 views

On convergence of convex bodies

Let $K\subset \mathbb{R}^n$ be a compact convex set of full dimension. Assume that $0\in \partial K$. Question 1. Is it true that there exists $\varepsilon_0>0$ such that for any ...
10
votes
1answer
146 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
0answers
28 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 ...
5
votes
1answer
115 views

Stochastic Covering Number of a Convex Set

Consider a convex set, say $S = [0,1]^d$. Let $X_1, X_2,\ldots,X_n, \ldots$ be i.i.d. random variables that are uniformly distributed on $S$. Denote the Euclidean ball centered at $x \in \mathbb{R}^2$ ...
4
votes
1answer
257 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
1answer
452 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 ...
1
vote
0answers
59 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 ...
18
votes
1answer
696 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
131 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
113 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
137 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. ...
1
vote
0answers
48 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 ...
1
vote
2answers
389 views

Terrain Generation: Infinite 2D space filled with Diffusion-limited aggregation clusters?

Disclaimer: I don't have a deep understanding of fractals or any higher math, I'm just personally interested in it, so please excuse me if I'm using wrong terms or if I'm being inaccurate. Making ...
14
votes
0answers
143 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 ...
17
votes
1answer
712 views

Infinite desert with waterpoints

UPDATE: I created a simple web-application, that allows the user to move waterpoints around, then automatically calculates a maximum set of interior-disjoint squares between the points (the algorithm ...
12
votes
2answers
256 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 ...
59
votes
18answers
6k views

One-step problems in geometry

I'm collecting advanced exercises in geometry. Ideally, each exercise should be solved by one trick and this trick should be useful elsewhere (say it gives an essential idea in some theory). If you ...
16
votes
1answer
396 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 ...
10
votes
1answer
227 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 ...
1
vote
1answer
117 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 ...
3
votes
0answers
165 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 ...
32
votes
14answers
7k views

Open problems in Euclidean geometry?

Which are some (research level) open problems in Euclidean geometry ? (Edit: I ask just out of curiosity, to understand how -and if- nowadays this is not a "dead" field yet) I should clarify a ...
4
votes
2answers
264 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 ...
1
vote
0answers
38 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
57 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 ...
9
votes
4answers
530 views

The distribution of the shortest path through $n$ points

In the big picture, I'd like to know: if I sample $n$ points uniformly at random in the unit square, what is the probability that the shortest path that visits each one of them is very small? More ...