**2**

votes

**0**answers

43 views

### Negative-curvature behaviour of higher-rank symmetric spaces

Let $X$ be a symmetric space of noncompact type with $rk\ X\geq 2$ and let $G$ be the identity component of the isometry group. Pick points $p\in X$ and $\xi\in\partial_{\infty}X$, with $\xi$ regular, ...

**2**

votes

**1**answer

89 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 $\lambda$-...

**6**

votes

**1**answer

145 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}{...

**4**

votes

**1**answer

152 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 $x\in X$, $0<r<\...

**3**

votes

**1**answer

114 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,\...

**0**

votes

**0**answers

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**

votes

**1**answer

159 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 ...

**4**

votes

**1**answer

121 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 $...

**15**

votes

**1**answer

610 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 $0<\...

**6**

votes

**1**answer

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

**1**

vote

**1**answer

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 $\vert x - y \vert \le R$,...

**13**

votes

**2**answers

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 ...

**4**

votes

**0**answers

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**

votes

**2**answers

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 ...

**22**

votes

**4**answers

1k views

### Visibility of vertices in polyhedra

Suppose $P$ is a closed polyhedron in space (i.e. a union of polygons which is homeomorphic to $S^2$) and $X$ is an interior point of $P$. Is it true that $X$ can see at least one vertex of $P$? More ...

**8**

votes

**1**answer

92 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. ...

**1**

vote

**1**answer

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**

votes

**1**answer

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}...

**2**

votes

**0**answers

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 ...

**4**

votes

**0**answers

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 ...

**15**

votes

**2**answers

603 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 ...

**5**

votes

**0**answers

68 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}}...

**1**

vote

**0**answers

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 ...

**7**

votes

**1**answer

128 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 ...

**6**

votes

**2**answers

320 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 \mathbb{N}$...

**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

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\...

**7**

votes

**3**answers

491 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

160 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 ...

**2**

votes

**0**answers

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**

vote

**1**answer

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**

votes

**3**answers

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**

votes

**1**answer

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**

votes

**3**answers

132 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.)...

**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 $(S^...

**7**

votes

**1**answer

261 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 $...

**7**

votes

**2**answers

570 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:
...

**10**

votes

**1**answer

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 ...

**10**

votes

**4**answers

760 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

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**

votes

**1**answer

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 ...

**9**

votes

**3**answers

785 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

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 ...

**5**

votes

**0**answers

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-...

**1**

vote

**0**answers

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 ...

**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

391 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

472 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 bit ...

**9**

votes

**2**answers

346 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 ...