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

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28 views

Area of a polyellipse

Pretty much like the title asks. I'll explain the situation. I have a set of points $N \in \mathbb{R^2}$. To select a next point $n_i$ to add to the set, select the node with the smallest combined ...
5
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1answer
156 views

CAT(0)-groups in dimension 2

Suppose I have a space $X$ which is connected, simply connected, CAT(0) of dimension 2 and a group $G$ which acts on $X$ freely, isometrically, properly discontinuously and cocompactly. What can be ...
6
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1answer
140 views

The radially symmetric isoperimetric problem

A solution to the $l$-isoperimetric problem on a Riemannian surface $(M,g)$ is a smooth closed curve $\gamma \subset M$ of length $l$ which minimizes the isoperimetric constant: $$h(\gamma) = \frac{l}{...
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1answer
50 views

Name for a uniform local boundedness property of a function

I am working with a function $f : \mathbb{R}^N \to \mathbb{R}$ having the property that for every $R > 0$, there exists $M > 0$ such that if $x, y \in \mathbb{R}^N$ and $\vert x - y \vert \le R$,...
4
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0answers
83 views

Yamaguchi submersion theorem

Let me remind first a theorem of Yamaguchi (1996). Given $n\in \mathbb{N}, \mu_0>0$. Then there exist $\delta_n>0$ and $\epsilon_n(\mu_0)>0$ with the following property. Let $X$ be an $n$-...
9
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2answers
465 views

Dissecting Ramanujan´s Cuboid: 1729 = 19 x 13 x 7

Consider the cuboid of dimensions 19 x 13 x 7 whose volume is 1729, the Hardy-Ramanujan number. What is the least number of smaller cuboids into which it can be dissected so that the resulting pieces ...
1
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1answer
51 views

Enclosing a convex plane domain in a disc

The following statement seems obvious to me: Let $\gamma:S^1\to\mathbb R^2$ be a smooth injection such that $\dot\gamma$ and $\ddot\gamma$ never vanish. Then $\gamma$ encloses a strictly convex ...
10
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1answer
377 views

The geometric median of a solid triangle

Let $\Omega\subset \mathbb R^n$ be a compact subset of dimension $n$. Define the geometric median on $\Omega$ as the point $m_{\Omega}\in \mathbb R^n$ such that the integral $\int_{\Omega}|x-m_{\Omega}...
8
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1answer
91 views

Differentiability of geodesics in Alexandrov subspaces of Riemannian manifolds

Let $M$ be a smooth Riemannian manifold. Let $X\subset M$ be a closed path connected subset which has curvature bounded below in the sense of Alexandrov with respect to the induced intrinsic metric. ...
2
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0answers
89 views

An elementary question about metrics on the real plane [closed]

Given the metric $d_p$ on the real plane, i.e. $$ d_p(x,y) = d_p((x_1, y_1), (x_2, y_2)) = [|x_1 - x_2|^p+ |y_1 - y_2|^p]^{1/p} $$ for which values of $p$ ($\geq 1$) is it true that the following ...
5
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67 views

Does the Hodge *-operator act on the tangent space at 0 to the space of integral (n-1)-cycles in a conformal manifold of dimension d=2n?

Suppose $M$ is a compact, oriented conformal manifold of even dimension $d=2n$. Write ${\cal D}^{\mathit{int}}_{k}(M)$ for the space of integral $k$-currents in $M$ and write ${\cal D}^{\mathit{int}}...
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0answers
149 views

Optimal instructions for the modular construction of rectlinear Lego structures

Let $X$ be a compact (or periodic) union of integer translates of unit cubes such that the interior of $X$ is connected. (If it makes any difference, suppose that the dimension $n$ of $X$ is 3.) I am ...
7
votes
1answer
126 views

A clarification on pointed Gromov-Hausdorff convergence

According to Burago-Burago-Ivanov, one says that the sequence of pointed metric spaces $(X_n,d_n,p_n)$ GH-converges to $(X,d,p)$ if for every $R>0,\varepsilon>0$, there exists a $N$ such that ...
3
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1answer
88 views

Sufficient conditions for a curve on the sphere to be the Gauß map of a closed path

I was wondering which curves on the $n-1$ sphere arise as the Gauss maps of closed paths in $\Bbb R^n$. Necessary conditions are obviously that the path on the sphere is the image of some smooth $S^1\...
2
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0answers
51 views

How much must a curve bend to intersect another curve twice?

Suppose $c_1$ and $c_2$ are segments of smooth plane curves. To be concrete, say $c_1$ and $c_2$ are graphs of smooth functions $f_i:[a_i,b_i]\to \mathbb R$, $i=1,2$. If the curves were lines, then ...
1
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1answer
81 views

Rigidity in a $CAT(-1)$ space

Summary: How to proove that a reunion of triangles in a $CAT(-1)$ space is isometric to the reunion of coresponding comparisons triangles ? Context and notations: Le $X$ be a $CAT(-1)$ metric space. ...
2
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3answers
126 views

What is the envelope formed by a triangle fixed to two points?

Take two fixed points in a plane and a triangle of fixed shape. Constrain two sides of the triangle to each touch one of the two points. As the triangle moves under this constraint the third side ...
0
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1answer
75 views

Showing convexity of a function in the unit ball

We have the unit sphere $S^2$ in $\mathbb{R}^3$ and two points, $X$ and $Y$ on the surface of the sphere. Then, a function is defined for any point $P$ inside of the unit ball as: $$f(P) = R\,d(P, XY)...
7
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2answers
568 views

Geometric or topological results from group theory

Do you know interesting examples of purely geometric or topological results which can be proved using group theory? To make precise what I have in mind, let us consider the two following examples: ...
4
votes
1answer
120 views

Comparison of angles in Alexandrov space

Let $X$ be a finite dimensional Alexandrov space with curvature bounded below. Let $p\in X$ be a fixed point. Is it true that for any $\varepsilon >0$ there exists $\delta>0$ such that for any $...
5
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3answers
345 views

Elementary reference for the isometry group of $\mathbb{RP}^2$

Endow the real projective plane with the distance defined by $d(L,L')$ := "the angle between the lines $L$ and $L'$ ". It is the case that every isometry from $RP^2$ onto $RP^2$ is induced by an ...
2
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1answer
167 views

Prove that a metric space is intrinsic

Let $(X,d)$ be a general locally compact metric space (in particular not a Riemannian manifold). Suppose we don't know if $(X,d)$ is complete. To prove $(X,d)$ is intrinsic. I have to compute the ...
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3answers
131 views

Ball ricochetting from a plane of close-packed spheres

Suppose the lower $z \le 0$ halfspace of $\mathbb{R}^3$ is filled with a rigid close-packing of unit-radius spheres. (I don't think it matters much for my purposes if it is an FCC or an HCP packing.)...
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0answers
103 views

Historical perspectives on CAT(0) spaces

Does there exist a survey on the early developments of CAT(k) spaces, with the first motivations and the first problems considered? I looked at Bridson and Haefliger's book On metric spaces of non-...
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0answers
109 views

Banach-Mazur distance from finite-dimensional subspaces of $\ell_p$ to the Hilbert space

I am reading a paper http://www.math.tamu.edu/~johnson/TF3.4.pdf by Bill Johnson and Andrzej Szankowski and having trouble grasping why $d_n(Z_m) \leq d_n(\ell_{p_{m+1}} ) = n^{|p_{m+1}-2|}$ in the ...
6
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2answers
159 views

Are rays in Carnot groups straight?

A famous open problem in Geometric Control Theory and in the study of sub-Riemannian manifolds is whether constant-speed length minimizers in a sub-Riemannian manifold are always smooth (see also this ...
5
votes
2answers
111 views

References for metrics in matrix groups

I am studying a very concrete matrix group with a riemaniann (right invariant) metric for solving a question on Applied Math. I need explicit formulas for the distance between two matrices, geodesics ...
13
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2answers
577 views

Lattice n-gons with ordered side lengths 1,2,3,…,n

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 ...
2
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1answer
82 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 ...
2
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1answer
92 views

A centralised website for computational attempts 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 ...
4
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0answers
67 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 ...
0
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1answer
81 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 ...
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
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4answers
326 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
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1answer
146 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 ...
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0answers
35 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
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1answer
107 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....
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3answers
471 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
131 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
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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
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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
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1answer
187 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
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1answer
64 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 ...
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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 ...
25
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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 ...
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1answer
741 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
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1answer
138 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
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0answers
119 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 ...
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0answers
59 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 ...