**13**

votes

**2**answers

509 views

### Geometric interpretation of exceptional Symmetric spaces

Elie Cartan has classified all compact symmetric spaces admitting a compact simple Lie group as their group of motion.There are 7 infinite series and 12 exceptional cases. The exceptional cases are ...

**6**

votes

**1**answer

98 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 ...

**2**

votes

**0**answers

63 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 ...

**6**

votes

**2**answers

316 views

### Motivation for Hirzebruch-Jung Modified Euclidean Algorithm

Let $a,b \in \mathbb{N} \ \ s.t. \ \ a > b$ have $\gcd(a,b) =1$. We can define the Hirzebruch-Jung modified euclidean algorithm as follows:
Let $e_i \in \mathbb{N} >2$, and $ r_k \in ...

**29**

votes

**5**answers

1k views

### Surfaces filled densely by a geodesic

Which smooth, closed surfaces $S \subset \mathbb{R}^3$ have no
single geodesic $\gamma$ that fills $S$ densely?
Say a geodesic $\gamma$ "fills $S$ densely" if the closure of the set of points
...

**3**

votes

**1**answer

83 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 ...

**7**

votes

**3**answers

485 views

### Self-Intersection of closed curves

Supoose I have a closed curve $\gamma$ in the plane such that for any isometry $g$ of $\mathbb{E}^2,$ such that $g(\gamma)\neq \gamma,$ $\gamma$ intersects $g(\gamma)$ in at most two points. It ...

**6**

votes

**2**answers

155 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 ...

**4**

votes

**1**answer

146 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

**1**answer

102 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 ...

**-3**

votes

**0**answers

64 views

### Volume of a scaled ball [closed]

Let $B \subset \mathbb{R}^n$ be the unit ball with respect to an arbitrary norm $\|.\|$ (e.g. $B=\{x \in \mathbb{R}^n:\|x\| \le 1 \}$).
I read in a book that it is easy to show:
$vol_n(\epsilon ...

**2**

votes

**0**answers

46 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**

vote

**1**answer

75 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**

votes

**3**answers

118 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**

votes

**1**answer

73 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, ...

**7**

votes

**3**answers

125 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 ...

**4**

votes

**1**answer

114 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 ...

**7**

votes

**2**answers

1k views

### Riemannian metrics as sections of a vector bundle

Let $\pi \colon E \to M$ be a smooth vector bundle. A Riemannian metric on $E$ can be regarded as a global section of the vector bundle $(E\otimes E)^{\ast}$, or more specifically, the subbundle ...

**7**

votes

**1**answer

259 views

### Is the center of gravity in a CAT(0) space contained in the convex hull?

In reading Greg Kuperberg's partial answer to this question Convex hull in CAT(0) ,
I started wondering if the center of gravity is always contained in the closed convex hull.
More precisely, given ...

**6**

votes

**2**answers

531 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:
...

**15**

votes

**1**answer

564 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

**1**answer

180 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 ...

**10**

votes

**4**answers

751 views

### In which geometries do triangles have an Euler line?

In Euclidean geometry, the centroid, orthocenter and circumcenter of a triangle lie on a line.
In which other geometries does this hold?

**5**

votes

**3**answers

342 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 ...

**5**

votes

**1**answer

133 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 $x \in \mathbb{R}^2$ ...

**2**

votes

**1**answer

159 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 ...

**9**

votes

**3**answers

783 views

### Can different bicycles leave the same tracks?

(asked by JST on the Q&A board at JMM)
Can two bicycles of different lengths leave the same set of tracks (aside from a straight line)?
[Ed: please tag this appropriately]

**2**

votes

**1**answer

91 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 ...

**12**

votes

**2**answers

504 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 ...

**5**

votes

**0**answers

92 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 ...

**1**

vote

**0**answers

101 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 ...

**41**

votes

**3**answers

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 ...

**14**

votes

**1**answer

367 views

### Generalizing the Mazur-Ulam theorem to convex sets with empty interior in Banach spaces

The Mazur-Ulam theorem (1932) states that any isometry of a normed linear space is affine. See Nica (Expo. Math. 30 (2012), 397-398; arXiv:1306.2380) for a very elegant proof.
Question: Let $M$ be a ...

**17**

votes

**3**answers

461 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 ...

**34**

votes

**14**answers

8k 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 ...

**9**

votes

**2**answers

340 views

### A question about the dispersion points of connected metric spaces

Let $C$ be an infinite, separable and connected metric space. If $C$ becomes totally disconnected when one of its points $p\in C$ is removed, does every closed ball of $C$ with
positive radius and ...

**5**

votes

**2**answers

104 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 ...

**2**

votes

**1**answer

81 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 ...

**8**

votes

**2**answers

906 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

**1**answer

79 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 ...

**2**

votes

**1**answer

81 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

**0**answers

66 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 ...

**4**

votes

**1**answer

144 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

**0**answers

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 ...

**11**

votes

**4**answers

321 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

**5**answers

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 ...

**1**

vote

**0**answers

31 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

**1**answer

246 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

**1**answer

104 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

**2**answers

129 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'. ...