**8**

votes

**0**answers

115 views

### Boomerangs in Polya's orchard

Polya's orchard problem asks for what radius $r$ of trees
at each lattice point within a distance $R$
of the origin block all lines of sight to the exterior of the orchard.
The answer is known; $r$ ...

**7**

votes

**1**answer

103 views

### When a Riemannian manifold with boundary is an Alexandrov space?

Let $M$ be a smooth Riemannian manifold (without boundary). Let $X\subset M$ be a smooth compact submanifold with boundary, $\dim X=\dim M$.
Under what conditions $X$, equipped with the induced ...

**1**

vote

**1**answer

62 views

### Existence or otherwise of a set of “sufficiently intricate” open cells

In my question Existence or otherwise of a set of "sufficiently intricate" open sets, I asked about whether it is possible to partition Lebesgue-almost all of $\mathbb{R}^d$ into a finite ...

**4**

votes

**1**answer

110 views

### Closest point to a dual lattice point (in particular for root lattices!)

Given a lattice $\Lambda\subset \mathbb{R}^n$ and a point $p\in\mathbb{R}^n$ outside the lattice, then I known it is a hard question to determine the set $S\subset \Lambda$ of all lattice points with ...

**3**

votes

**1**answer

98 views

### Existence or otherwise of a set of “sufficiently intricate” open sets

Fix $d \in \mathbb{N}$. Do there exist mutually disjoint connected open sets $V_1,\ldots,V_n \subset \mathbb{R}^d$ and $\mathbf{v} \in \mathbb{R}^d$ such that
$\mathbb{R}^d \setminus ...

**5**

votes

**0**answers

68 views

### What is the maximal convex hull in $\mathbb R^3$ of a tree with fixed total length?

Denote by $\mathcal T_n$ the set of all trees on $n$ nodes. For a tree $T\in\mathcal T_n$, we assign to each edge a non-negative length such that the sum of all lengths is 1. Denote by $v(T)$ the ...

**4**

votes

**0**answers

102 views

### Volume growth of balls

Let $G$ be a locally compact group and $K\subset G$ a compact subgroup. Suppose that on the homogeneous space $X=G/K$ we have a $G$-invariant proper metric $d$. For $R>0$ let $B(R)$ be the open ...

**1**

vote

**1**answer

89 views

### Lamination as limit of arcs

I am reading Bonahon's notes on closed curves, in particular the part about hyperbolic laminations. In his notes Bonahon illustrates some examples as why laminations should be "limit curves" on ...

**4**

votes

**1**answer

121 views

### Classification of 2-dimensional Alexandrov spaces

Is it possible to classify explicitly compact 2-dimensional Alexandrov spaces with curvature bounded below (either with or without boundary)?
If yes, a reference would be helpful.
EDIT: If the ...

**5**

votes

**2**answers

254 views

### Which surfaces admit unbounded-length simple geodesics?

Let $S$ be a surface embedded in $\mathbb{R}^3$.
A simple geodesic on $S$ is one that does not self-intersect.
Some surfaces have simple geodesics whose length exceeds any
given bound $L$. For ...

**3**

votes

**0**answers

157 views

### Distance between quadratic forms

In notes here http://math.univ-lyon1.fr/homes-www/gille/prenotes/lens.pdf on page $2$ a formulation of distance between two positive quadratic form $[q],[q']$ is given by
...

**5**

votes

**1**answer

105 views

### Geometry of convex subsets in Alexandrov space/ Riemannian manifold

Let $X^n$ be an $n$-dimensional complete Alexandrov space with curvature bounded below (or a smooth Riemannian manifold, possibly with boundary). Let $U\subset X$ be an open dense subset with the ...

**5**

votes

**1**answer

131 views

### Are the primary parallelotopes classified? (equivalently, Voronoi cells of lattices)

A primary parallelohedron is a polyhedron that can fill space with infinite translated copies.
It is known (e.g., Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 29-30, 1973; or, ...

**1**

vote

**1**answer

68 views

### Optimal covering with finite subcollection of open sets

This is mainly a reference request. Consider a finite collection of (let's say, for simplicity) of open balls $B_i, i = 1, 2, ..., m$ in (again, for simplicity) $\mathbb{R}^n$. I am looking for ...

**4**

votes

**2**answers

170 views

### Volume of the convex hull of the set of all graphic sequences of a given length

Consider the set of all graphic sequences with $n$ elements as a subset of $\mathbb{R}^{n}$, namely let
$$D(n)=\{(d_{1},\dots,d_{n})\in\mathbb{Z}_{+}^{n}:d_{1}\geq\dots\geq d_{n},\ ...

**7**

votes

**1**answer

71 views

### Convergence of functions on Alexandrov spaces

Consider a sequence of $n-$dim Alexandrov spaces with curvature $\geq$ -1 $\{(M_i,p_i)\}$ Gromov-Hausdroff converging to an $n-$dim Alexandrov space $(M,p)$. Let $f:M\mapsto \mathbb R$ be a Lipschitz ...

**2**

votes

**1**answer

63 views

### A property of concave functions on Alexandrov spaces

EDIT: Let $X$ be an $n$-dimensional Alexandrov space with curvature bounded below. Let $f_1,\dots, f_n\colon X\to \mathbb{R}$ be $\lambda$-concave functions. Assume that at a fixed point $p$ there ...

**0**

votes

**1**answer

58 views

### A bound on the Haussdorff distance

Let $X, Y \subset \mathbb{Z}^2$ be two discrete and bounded sets. Let $f_X$ be the Euclidean signed distance function of $X$ (similarly for $Y$) and $d_H(X,Y)$ the Euclidean Haussdorff distance ...

**3**

votes

**1**answer

96 views

### Reference: Finsler Derivative?

On the wikipedia page "Generalizations of derivative" the author mentions: " in Finsler geometry, one studies spaces which look locally like Banach spaces. Thus one might want a derivative with some ...

**6**

votes

**1**answer

93 views

### Countable subcover of half-open cylinders

While preparing a lecture on dynamic programming principle in optimal stochastic control after the book of Touzi, I discovered a gap in the proof of DPP (page 28 of the book).
Here I simplify the ...

**1**

vote

**2**answers

191 views

### Geodesic on Banach Manifold [closed]

Is there a way of defining a geodesic on a Banach Manifold $M$ which is not itself a Hilbert Manifold?

**2**

votes

**1**answer

96 views

### Why simple closed curves are dense in $\mathcal{PML}_0(S)$?

I have another question about laminations on surfaces. As usual let $\mathcal{S}$ be the set of homotopy classes of simple closed curves in $S$ and $\mathcal{PML}_0(S)$ be the set of projective ...

**2**

votes

**0**answers

64 views

### Shortest paths stepping on rational points of height $h$

Q. Do shortest paths walking between rational points of height $h$
ever properly cross themselves?
Explaining this question takes a bit of definitional exposition.
First, I copy definitions ...

**3**

votes

**1**answer

120 views

### Why is $\mathcal{PML}_0(S)$ compact?

I'm starting to study geodesic laminations on hyperbolic surfaces and in particular I'm focusing my attention on $\mathcal{PML}_0(S)$, the space of projective classes of measured geodesic laminations ...

**2**

votes

**1**answer

62 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 ...

**11**

votes

**1**answer

387 views

### Strong equivalence between intrinsic and extrinsic metrics on $GL_n^+$?

$\newcommand{\til}{\tilde}$
Lately, I have become interested in comparing intrinsic and extrinsic metrics on Riemannian manifolds.
Consider $GL_n^+$ (invertible matrices , $\det >0$) as an open ...

**14**

votes

**2**answers

555 views

### Algebraic surface of constant width?

Does there exist an irreducible polynomial $f \in \mathbb{R}[x, y, z]$ such that:
$$ V := \{ (x, y, z) \in \mathbb{R}^3 : f(x, y, z) \leq 0 \} $$
is a solid of constant width with a finite symmetry ...

**3**

votes

**0**answers

59 views

### Equidistribution of Brillouin zones

Answering the question about Limiting shape for Brillouin zones Victor Kleptsyn proved that $N$th Brillouin zone is very close to a circle of radius $c\sqrt N$ (you can find all necessary definitions ...

**4**

votes

**1**answer

80 views

### Gromov-Hausdroff convergence for Alexandrov spaces

Let $\{X_n\}_{n=1}^\infty$ be a sequence of compact Alexandrov spaces (with curvature $\geq k$) converging to (in the sense of Gromov-Hausdroff convergence) an Alexandrov spaces $X$, and ...

**8**

votes

**2**answers

290 views

### Constructing a function over a metric space through given points

Suppose there is a compact metric space $(X,\rho)$ and a Euclidean space $\mathbb{R}^n$.
There is a sequence of unequal points $\{x_1,...x_N\}$ in $X$ such that all metrics $\rho(x_i,x_j)$ are known ...

**11**

votes

**0**answers

114 views

### GPS calculations under $L^p$ norms

GPS calculations require finding a sphere externally tangent to
four given spheres, an
Apollonian problem
in $\mathbb{R}^3$.
The center of that fifth sphere is one of the $16$ possible solutions to
...

**4**

votes

**1**answer

198 views

### Combinatorial description of a 120-cell

I'd like a combinatorial description of the 1-skeleton of the 120-cell (roughly) along the lines of the following description of the 1-skeleton of a dodecahedron.
(View all elements of product sets ...

**5**

votes

**0**answers

108 views

### Connectedness of cones in the boundary of a 1-ended hyperbolic group

Let $G$ be a one-ended hyperbolic group. We can think of the boundary of $G$ as consisting of geodesic rays originating at the identity in some Cayley graph, modulo the relationship of being ...

**6**

votes

**1**answer

115 views

### Self-avoiding/reflecting geodesics on a convex surface

Let $S$ be the surface of a convex body embedded in $\mathbb{R}^3$.
For me $S$ is a convex polyhedron,
but I am happy to view $S$ as a smooth body with positive Gaussian curvature
at each point, or ...

**12**

votes

**1**answer

299 views

### Limiting shape for Brillouin zones

Is it true that the limiting shape for Brillouin zones (for any lattice) is a circle?
You can find the definition and the step by step construction of Brillouin zones here. This picture is taken from ...

**29**

votes

**5**answers

1k views

### Tiling the plane with incongruent isosceles triangles

It is not difficult to tile the plane with incongruent triangles.
One could tile with equilateral triangles, and then partition
each equilateral into three triangles, displacing their common
...

**5**

votes

**1**answer

174 views

### Convergence in the proof of Crofton's Formula

Let $\mathcal{L}$ be the set of oriented lines in $\mathbb{R}^2$ and let $\mu$ be the Kinematic Measure on $\mathcal{L}$; up to scaling, $\mu$ comes from the unique (up to scaling) volume form on ...

**24**

votes

**0**answers

397 views

### Isometric embeddings of finite subsets of $\ell_2$ into infinite-dimensional Banach spaces

Question: Does there exist a finite subset $F$ of $\ell_2$ and an infinite-dimensional Banach space $X$ such that $F$ does not admit an isometric embedding into $X$?
There are some results of the ...

**5**

votes

**1**answer

154 views

### Intersection of rotating regular polygons

This question has a recreational flavor, but may not be
entirely uninteresting.
Let $P_k$ be a unit-radius regular polygon of $k$ sides,
and $P_n$ a unit-radius regular polygon of $n \ge k$ sides.
...

**5**

votes

**1**answer

73 views

### Determining the stretch of a cluster of points

I am trying to determine a metric for measuring cluster stretch. Let C be a cluster of points P0,P1,...,Pn in a two dimensional space with same units.
I need a metric that will allow me to ...

**6**

votes

**2**answers

275 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 ...

**6**

votes

**3**answers

395 views

### Defining Euler's number via elementary euclidean geometry (and a dimension limit)

Let $B_n$ be a closed ball in euclidean space $\mathbb{R}^n$, and consider the largest cube $Q_n$ contained in $B_n$. Then, let $C_n$ be a cube of maximal size that is contained in $B_n$ and disjoint ...

**4**

votes

**0**answers

78 views

### Finding closest set of K disjoint hyperspheres to a point in $\mathbb{R}^n$ with uniform radius

I am interested in the following problem: in $\mathbb{R}^n$, we have $N$ overlapping hyperspheres all with the same radius. Given a point $p$ in $\mathbb{R}^n$, the objective is to find the $K$ non ...

**7**

votes

**2**answers

301 views

### Closed curve whose neighborhood is as large as possible

Let $C$ be a closed curve in the plane and let $N_\epsilon(C)$ be an $\epsilon$-neighborhood of $C$, like this:
(ignore the fact that the "curve" is polygonal in this picture, I drew it in MATLAB)
...

**1**

vote

**0**answers

71 views

### Of the standard distance metrics, which ones can/cannot be embedded in Euclidean space?

Given the discussion from:
Representability of finite metric spaces
it appears that a 1974 paper by Morgan gives the criteria for when a distance metric can be embedded in Euclidean space. My first ...

**7**

votes

**2**answers

283 views

### Does the hyperdeterminant calculate a quantity akin to the volume of a parallelepiped?

If $M$ is an $n \times n$ matrix, $|\det(M)|$ is the volume of the $n$-dimensional
parallelepiped spanned by the column vectors of $M$.
...

**4**

votes

**1**answer

69 views

### Quantitative version of the splitting theorem

The classical splitting theorem (Toponogov, Milka) says that if a smooth complete Riemannian manifold (more generally, Alexandrov space) $M^n$ of non-negative sectional curvature contains a line (i.e. ...

**3**

votes

**0**answers

141 views

### ricci flow on surfaces

In Hamiltons paper "Ricci flow on surfaces" there is an estimate on $|\nabla R|^2$ which shows that $|\nabla R|^2 \leq C_1 \exp{\frac{rt}{2}}$ for some constant $C_1$.
Actually for any solution of the ...

**13**

votes

**0**answers

95 views

### A kaleidoscopic coloring of the plane

Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap ...

**6**

votes

**1**answer

130 views

### Existence of a measurable map between metric spaces

Let $X$ and $Y$ be separable complete metric spaces (if necessary, they may be assumed to be compact). Let $R\subset X\times Y$ be a closed subset such that the projection of $R$ to $X$ is onto.
Is ...