**3**

votes

**1**answer

147 views

### How many points are in such set with the same norm-2

Let $L=[a,b]\cap\mathbb{N}$ with $a,b\in\mathbb{N}$, let $D\in\mathbb{N}$, and let $C=L^D$. Then I would like to know how many points are there in $C$ with the same given norm-2 $d$. I.e., I'm looking ...

**3**

votes

**1**answer

125 views

### How to show it is contained in a convex hull?

There are $(d+1)f$ points (denote the set of all points as $S$) in $\mathbb{R}^d$, that can be divide into $d+1$ disjoint sets $F_1,...,F_{d+1}$, each set of size $f$. If we have
$$
...

**8**

votes

**2**answers

300 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

119 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

201 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

111 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

118 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

339 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 ...

**4**

votes

**1**answer

124 views

### what's the formula of the inradius of a general simplex? [closed]

As the title, I just want to know whether there is a general formula for calculating the inradius of a n-simplex. Thank you!

**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

183 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

477 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

160 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

78 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

317 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

401 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

80 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

307 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)
...

**25**

votes

**0**answers

521 views

### Tiling a square with rectangles

Is it possible to completely tile a square with different rectangles of integer sides but all with the same area?
The original problem, not requiring integer sides for rectangles, was proposed by Joe ...

**1**

vote

**0**answers

77 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 ...

**8**

votes

**2**answers

330 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

71 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

149 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 ...

**14**

votes

**0**answers

104 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

136 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 ...

**3**

votes

**1**answer

65 views

### Is the boundary of Alexandrov space again an Alexandrov space?

Let $X$ be a finite dimensional (possibly compact) Alexandrov space with curvature $\geq K$. Is it true that its boundary is again Alexandrov space with curvature bounded from below? If yes, is the ...

**6**

votes

**2**answers

103 views

### Equality of triangles in normed spaces

Hilbert space satisfies the following condition: if two triangles $\triangle ABC$, $\triangle A_1B_1C_1$ have equal sides lengths: $|AB|=|A_1B_1|$, $|BC|=|B_1C_1|$, $|AC|=|A_1C_1|$ they also have ...

**12**

votes

**2**answers

425 views

### Embedding Z into Z^2 with large distortion

Is it possible to find a 2-way infinite (self-avoiding) path $\{x_i\}_{i\in \mathbb Z}$ in the standard Cayley graph of $\mathbb Z^2$, i.e. the square grid, such that the distance between $x_i$ and ...

**2**

votes

**1**answer

77 views

### A property of geodesic triangles in Alexandrov spaces

Let $X$ be an $n$-dimensional Alexandrov space with curvature at least -1. Assume that at every point it has an $(n,\delta)$-strainer of length $\mu$, where $\delta$ and $\mu$ are independent of a ...

**7**

votes

**1**answer

319 views

### Area of square to wrap a torus

The Nash-Kuiper
$C^1$ isometric embedding of flat torus into $\mathbb{R}^3$
has recently been spectacularly visualized by the
Hevea Project.
This suggests two questions.
Q1. What is the area of ...

**7**

votes

**0**answers

129 views

### Bang's open question strengthening Tarski's planks problem

Tarski's Planks problem,
solved by Thøger Bang in 1951, says (in a simplified $\mathbb{R}^2$ version) that it requires
"planks" (parallel strips) of total width $\ge d$ in order to completely cover
a ...

**2**

votes

**0**answers

91 views

### What is the metric on the Fuchsian model? [closed]

Let $\mathbb{H}$ be the upper half plane, and $\Gamma < SL(2, \mathbb{R})$ be a Fuchsian group. How is the distance between any two points $x, y \in \mathbb{H} / \Gamma$ in the Fuchsian model ...

**10**

votes

**2**answers

316 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 ...

**4**

votes

**1**answer

90 views

### A property of geodesic triangles in manifolds with lower bounds on curvature and injectivity radius

Does there exist a function $\tau(\varepsilon)=\tau(\varepsilon,n,K,\mu)$ such that $\lim_{\varepsilon\to +0}\tau(\varepsilon)=0$ and for any $n$-dimensional complete Riemannian manifold $M^n$ with ...

**4**

votes

**0**answers

88 views

### Sharp isoperimetry in the discrete Heisenberg group

The exact shape of the set which has the best isoperimetry in the continuous Heisenberg is (from what I know) a difficult open problem. This brought to wonder what is known in the discrete case?
More ...

**3**

votes

**1**answer

120 views

### Lebesgue differentiation theorem holds on locally doubling space?

It's known that for a metric space with doubling measure $(X,\mu)$, the Lebesgue differentiation theorem holds , i.e. If $f:X\to \mathbb{R}$ is a locally integrable function, then $\mu$-a.e. points ...

**1**

vote

**1**answer

31 views

### Multiplicity of a subcovering in spaces of given Hausdorff dimension

Let $X$ be a locally compact metric space of integer Hausdorff dimension $n$. Let $K\subset X$ be a compact subset. Let $\{B_i\}_i$ be a finite family of balls covering $K$. One may assume that all ...

**4**

votes

**1**answer

132 views

### local quasi geodesics in hyperbolic spaces

I asked this question on math stackexchange (see here) but didn't get any answer so I thought I post it here too.
We have the following two well-known Theorems:
T1) For all $\delta > 0, ...

**2**

votes

**1**answer

175 views

### Is it possible to construct an isosceles triangle by using a ruler and without using a pair of compasses?

It is well known that on Euclidean plane one can construct an isosceles triangle on given straight line by using a ruler and a pair of compasses.
Also it is possible to construct straight line ...

**1**

vote

**1**answer

85 views

### Pairs of rays in euclidean buildings

In section 4.1.3 of Kleiner and Leeb's paper on symmetric spaces and euclidean buildings, there's a result about pairs of rays from the same point initially spanning a flat triangle (or being ...

**8**

votes

**1**answer

178 views

### $|\exp_p(x)\exp_q(T(x))|$ controlled by $|pq|?$ $T$ is parallel transportation in Alexandrov space

Let $M$ be an Alexandrov space with $sec \geqslant 0$. Let $p$ and $q$ be points of shortest path $\gamma$ in $M$, that are not end point. Then the tangent cone can split as $C_p=L_p \times ...

**1**

vote

**1**answer

252 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 ...

**12**

votes

**3**answers

279 views

### Curvature of a finite metric space

I am sorry to ask a very vague question, but:
What are good ways to define the curvature of a finite metric space?
The best way I can think of is: the curvature of a finite metric space $M$
is ...

**5**

votes

**1**answer

78 views

### Volume satisfying inequality constraints (simplex subset)

Is there a way to find the volume of the "feasible region" of a standard simplex satisfying simple range constraints?
$x_1+x_2+...+x_n = 1$
$a_1 \le x_1 \le b_1$
$a_2 \le x_2 \le b_2$
$...$
$a_n \le ...

**2**

votes

**1**answer

113 views

### Choosing the weights of a Voronoi diagram — is this function always the gradient of another function?

This question is related to the earlier question Weighted area of a Voronoi cell . As in that question, let $X = \{ x_1,\dots,x_n\} $ denote a set of $n$ points in the unit square $S = ...

**3**

votes

**2**answers

89 views

### Basic question about discrete minimal surfaces

Let $P$ be a convex polygon with $n > 3$ vertices $v_1, \ldots, v_n \in \mathbb{R}^2$, let $x$ be a point in the interior of $P$, and let $u$ be a function with prescribed values at the vertices of ...

**1**

vote

**0**answers

79 views

### A compact Alexandrov space with curvature bounded below has curvature bouneded above? [closed]

For a compact Riemannian manifold, Since the curvature tensor is continuous, we know that the sectional curvature is bounded, i.e. bounded above and below. Now let $M$ be a compact Alexandrov space ...

**9**

votes

**2**answers

343 views

### For a 3-manifold $Y$, when does $Y\times S^{1}$ admits a Riemannian metric with positive scalar curvature?

Let $Y$ be an orientable, smooth 3-manifold and let $X=Y\times S^{1}$. My question is that: when does $X$ admits a Riemannian metric with positive scalar curvature?
An obvious case is when $Y$ ...

**6**

votes

**2**answers

453 views

### Geodesics on SO(3)

I have two 3D rotations about the origin, represented as
$3 \times 3$ orthogonal matrices $M_1$ and $M_2$
(specified by numerical entries),
and I would like to interpolate (and compute)
a continuous ...

**4**

votes

**0**answers

112 views

### Are ultralimits the Gromov-Hausdorff limits of a subsequence?

Let $(M_i,p_i)$ be a sequence of $n$-dimensional Riemannian manifolds with lower Ricci curvature bound $-1$. Fix a non-orincipal ultrafilter and let X be the ultralimit of the sequence.
Does there ...