**21**

votes

**0**answers

269 views

+50

### 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 ...

**-1**

votes

**0**answers

38 views

### Alternative Geometries [migrated]

In our world, the distance between two points (in 2d) is defined as $\sqrt{(\Delta x)^2 + (\Delta y)^2}$. Suppose that in an alternative geometry, it was defined as $\sqrt[p]{|\Delta x|^p + |\Delta ...

**4**

votes

**1**answer

98 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 ...

**7**

votes

**1**answer

130 views

### Generalization of Stewart's theorem?

I'm curious about the generalization of Stewart's theorem to more dimensions. MathWorld mentions that there is a generalization done by Bottema, but I could not find much information on it. All I ...

**12**

votes

**1**answer

201 views

+50

### bi-Lipschitz gluing

Let $H$ be the Heisenberg group with
left invariant sub-Riemannian metric and $\varepsilon>0$ is small.
Let us denote by $|x-y|_H$ the distance from $x$ to $y$ in $H$.
I have a bi-Lipschitz ...

**2**

votes

**1**answer

99 views

### a property implying co-circularity [closed]

Let $A_1, A_2,\ldots, A_n$ be $n$ distinct points in the plane.
For every $1\le i\le n$, let $D_i$ be the sum of the distances from point $A_i$ to all the other points.
Suppose that $D_i=D_j$ for ...

**77**

votes

**5**answers

3k views

### Light rays bouncing in twisted tubes

Imagine a smooth curve $c$ sweeping out a unit-radius disk that is
orthogonal to the curve at every point.
Call the result a tube.
I want to restrict the radius of curvature of $c$ to be at most 1.
I ...

**0**

votes

**0**answers

40 views

### What is the minimal number of lines needed to partition a simplex into cells of diameter at most $\epsilon$?

I am studying a problem that requires me to partition the simplex into cells using a particular family of hyperplanes. For concreteness, consider the 2-simplex. I would like to construct lines ...

**6**

votes

**1**answer

74 views

### metric condition forcing convex position

Let $A_1,A_2,\ldots, A_n$ be distinct in the plane. For every $1\le i \le n$, let
$S_i=\sum\limits_{j=1}^n d(A_i,A_j)$ be the sum of distances from $A_i$ to all the other points.
Assume that ...

**7**

votes

**0**answers

100 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; $r$ ...

**16**

votes

**5**answers

1k views

### Definition of area

I am looking for an attractive, but rigorous definition of area;
say in Euclidean plane. Probably there is no short definition. It is OK to make it even longer, but can it be built from useful parts ...

**1**

vote

**1**answer

58 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 ...

**7**

votes

**1**answer

86 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 ...

**2**

votes

**1**answer

88 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

**1**answer

126 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, ...

**0**

votes

**1**answer

151 views

### Generalized Sphere Kissing Problem [duplicate]

I am currently researching discrete geometry and I am in need of an upper bound on a generalized kissing number in 3-dimensions dependent upon a parameter $\eta$ which is the radii of spheres touching ...

**12**

votes

**1**answer

306 views

### Walls of CAT(0) cube complex sufficiently far apart implies intersection of stabilizers finite

I was reading through Agol's paper on the Virtual Haken Conjecture and I came across a claim whose proof I am after. It seems to boil down to the following claim about the hyperplanes and their ...

**7**

votes

**1**answer

226 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 ...

**4**

votes

**1**answer

175 views

### when are local quasigeodesics global in CAT(0)

It is well-known (and easily shown) that a local quasi-geodesic (for some value of "local") in a $\delta$-hyperbolic space is global (one can compute the constants, as well, from local data). This is ...

**1**

vote

**1**answer

123 views

### Convex subcomplexes of CAT(0) cubical complexes

Is the following statement true? If so, can anyone provide a reference?
Let $X$ be a CAT(0) cubical complex, and let $Y$ be a connected
subcomplex of $X$. Then the following are equivalent:
...

**6**

votes

**0**answers

143 views

### CAT(0) groups that does not act on CAT(0) cubical complex

CAT(0) groups are groups that act on a CAT(0) space properly and cocompactly. If a group acts on a CAT(0) cubical complex properly and cocompactly, then of course it is a CAT(0) Group. I am wondering ...

**13**

votes

**1**answer

395 views

### Mapping class group and CAT(0) spaces

I hope the questions are not too vague.
Is the mapping class group of an orientable punctured surface $CAT(0)$ ?
Is any of the remarkable simplicial complexes (curve complex, arc complex...) built ...

**2**

votes

**1**answer

84 views

### Is a cocompact CAT(0) periodic?

Let $X$ be a CAT(0) space and $G$ its group of isometries. Then $X$ is said to be cocompact, if there exists a compact set $K\subset X$ with $X=G.K$. The space $X$ is called periodic, if there exists ...

**8**

votes

**2**answers

295 views

### Isometric imbedding of a sphere with positively curved metric

QUESTION. Given a Riemannian metric on the sphere $S^n$ with positive sectional survature. Can it be isometrically imbedded into $\mathbb{R}^{n+1}$ (of any class of regularity) as a boundary of a ...

**5**

votes

**0**answers

65 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

97 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

72 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 ...

**70**

votes

**11**answers

9k views

### Is it possible to capture a sphere in a knot?

You and I decide to play a game:
To start off with, I provide you with a frictionless, perfectly spherical sphere, along with a frictionless, unstretchable, infinitely thin magical rope. This rope ...

**4**

votes

**1**answer

99 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

239 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 ...

**7**

votes

**2**answers

186 views

### Can every large point set be connected to a given knot?

Let $K$ be a given knot, and
$P$ a set of points in $\mathbb{R}^3$ in general position,
general position in the sense that no three points are collinear
and no four coplanar.
Define the point-set ...

**3**

votes

**0**answers

148 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
...

**6**

votes

**1**answer

92 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 ...

**5**

votes

**1**answer

98 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 ...

**0**

votes

**0**answers

83 views

### Hausdorff dimension of homeomorphic compact metric spaces [migrated]

1) Are there examples of homeomorphic compact metric spaces of different Hausdorff dimension?
2) If yes, are there sufficient conditions on the spaces which would imply the equality of Hausdorff ...

**1**

vote

**1**answer

61 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 ...

**24**

votes

**2**answers

1k views

### Tiling of the plane with manholes

Some shapes, such as the disk or the Releaux triangle can be used as manholes,
that is, it is a curve of constant width.
(The width between two parallel tangents to the curve are independent of the ...

**7**

votes

**1**answer

65 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

49 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 ...

**2**

votes

**0**answers

60 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

126 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},\ ...

**1**

vote

**2**answers

183 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?

**0**

votes

**1**answer

57 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

86 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 ...

**50**

votes

**1**answer

642 views

### Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$

Let $f : \mathbb{R} \longrightarrow \mathbb{R}^2$ be a continuous map which sends any interval $I \subseteq \mathbb{R}$ to a convex subset $f(I)$ of $\mathbb{R}^2$. Is it true that there must be a ...

**0**

votes

**1**answer

176 views

### Is the map $\exp_x(\nabla_x \sum_{i=1}^m d^2(x_i,x))$ Lipschitz?

The last question is too general, this is a modification.
Let $M$ be an $n$ dimensional Riemannian manifold. Fix $m$ points $x_1,...,x_m$. Suppose $y$ is not in the cut locus of $x_i$ for $1 ...

**14**

votes

**2**answers

535 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 ...

**2**

votes

**1**answer

91 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 ...

**7**

votes

**7**answers

2k views

### Uniformly Sampling from Convex Polytopes

How to choose a point uniformly from a convex polytope $P \subset [0,1]^n$ defined by some inequalities, Ax < b ? (Here A is an m-by-n matrix, x is n-by-1 and b is m-by-1.) I imagine that you ...

**3**

votes

**1**answer

114 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 ...