# Tagged Questions

**9**

votes

**0**answers

144 views

### Characterizing the norms on $\mathbb{R}^3$ coming from Platonic solids

Recall that any sufficiently nice compact centrally symmetric convex body in $B \subset \mathbb{R}^3$ gives rise to a Banach norm on $\mathbb{R}^3$ which has $B$ as its unit ball.
Is there a nice ...

**3**

votes

**2**answers

307 views

### Reference request for an early theorem of Gromov

In his talk Misha Gromov- How does he do it, Jeff Cheeger mentions a theorem of Gromov proved sometime in the early 70's. Theorem: Every manifold admitting a sequence of metrics such that the diameter ...

**13**

votes

**4**answers

802 views

### What is the analog of the “Fundamental Theorem of Space Curves,” for surfaces, and beyond?

The "Fundamental Theorem of Space Curves"
(Wikipedia link; MathWorld link)
states that there is a unique (up to congruence)
curve in space that simultaneously realizes
given continuous curvature ...

**1**

vote

**1**answer

102 views

### Mahler Volume of the Snub 24-Cell

I have been working recently on the Mahler conjecture and I am interested in what the Mahler volume of the snub 24-cell in order to check an example calculation.
The recent paper regarding the snub ...

**5**

votes

**1**answer

305 views

### Background to understand Gromov's green book

I have a decent background in differential geometry. I have read John Lee's introduction to smooth manifolds and doCarmo's Riemannian Geometry. I was trying to read Misha Gromov's Metric structures ...

**2**

votes

**0**answers

60 views

### A cylinder leaking Brownian particles, cut in half by a mirror

This question is tangentially related to Probability a Brownian particle with an exponentially distributed lifetime hits a sphere before vanishing.
I have an infinitely long cylinder of some radius ...

**2**

votes

**1**answer

150 views

### Distance measure for noisy $SE(3)$ transforms

I have a transformation $T \in SE(3)$ parameterized by a mean quaternion $q$ with covariance matrix $\Sigma_q \in R^{4\times4}$ and a mean translation $t \in R^3$ with covariance matrix $\Sigma_t \in ...

**1**

vote

**0**answers

89 views

### On 'Very Movable' Geometric Configurations (Configurations with a large degree of freedom)

Let $C$ be an $(n_r, b_k)$ combinatorial configuration that admits a geometric realization in the plane. I'm interested in the maximum number of points/lines $M$ of $C$ we can place in general ...

**1**

vote

**2**answers

365 views

### Isometric embeddings of metric spaces in Hilbert spaces

There are plenty of isometric embeddings of metric spaces in Banach spaces. Nevertheless, I have been unable to find any significant result on isometric embeddings into Hilbert spaces. My question is: ...

**2**

votes

**0**answers

82 views

### Convex polyhedra jammed in $k$ disjoint holes

For a given convex polyhedron $P \subset \mathbb{R}^3$,
I was imagining finding the optimal "fixing" of $P$ in holes (or jamming them in "mud"),
which led to the following question.
First, scale $P$ ...

**1**

vote

**1**answer

194 views

### Normal tubular neighborhood theorem for semi(or pseudo)-riemannian manifolds

Suppose you have a manifold $M$ and a closed sub-manifold $A$, and let $g$ be a semi-riemannian metric,ie, $g_x$ defines a quadratic form on $T_xM$ such that $g_x(v,v)\ge0$, but $g_x(v,v)=0$ not ...

**22**

votes

**3**answers

2k views

### Can a unit square be cut into rectangles that tile a rectangle with irrational sides?

For arbitrary positive integers $m$ and $n$, if we dissect a unit square into an $m\times n$ rectangular grid of $1/m\times 1/n$ rectangles, we can reassemble these $mn$ rectangles into an $n/m\times ...

**5**

votes

**1**answer

175 views

### Simultaneous geometric separator

A geometric separator is a line that separates a given set of shapes to two subsets of approximately the same size (up to a constant), while intersecting only a small number of shapes. When a ...

**3**

votes

**1**answer

115 views

### Is any finite collection of points contained in a cut and project set with $\mathbb{R}^d$ as internal space?

In "Meyer Sets and their Duals" Moody proves that any Meyer set union a finite number of points is again a Meyer set. Additionally, any Meyer set is contained in a finite union of model sets whose ...

**7**

votes

**2**answers

316 views

### $\mathrm{Bessel}^3$ Integral

I'm trying to calculate the following integral:
$\int_0^\infty \mathrm{BesselJ}[l_0,k_0r] \cdot \mathrm{BesselJ}[l_1,k_1r] \cdot \mathrm{BesselJ}[l_0-l_1,kr] \cdot r\,dr$
($\mathrm{BesselJ}[n,x]$ is ...

**1**

vote

**1**answer

202 views

### Harmonic function defined on a cone

It's well known that: Given a continuous function defined on the boundary of the disk, then there exists a unique harmonic function in the interior of the disk. What if we replace the disk by a cone?
...

**19**

votes

**5**answers

805 views

### Lightray trapped between two mirror disks: Computation formulation?

I would like to calculate the angle of a ray $r$ from a given
point $p$ such that it gets "stuck" reflecting between
two congruent mirror-disks.
For why there is such a ray, see the (amazing!) answer
...

**0**

votes

**1**answer

119 views

### What is the area of the piece of an $n$-sphere within a given angle of a vector? [closed]

Let $x$ be the unit vector $(1,0,0,\ldots,0)$ in $\mathbb{R}^n$, and let $A(\theta)$ be the subset of $\mathcal{S}^{n-1}$ whose angle to $x$ is less than $\theta$, i.e.
$$ A(\theta) = \left\{ y \in ...

**9**

votes

**1**answer

196 views

### Is it always possible to “encircle” exactly $n$ points in an infinite subset of $\mathbb{R}^d$ without limit points?

Let $d$ be a positive integer, and let $\mathbb{R}^d$ be endowed with the
Euclidean metric. Given an infinite set $S \subset \mathbb{R}^d$ without limit points
and a positive integer $n$, is there ...

**3**

votes

**1**answer

80 views

### Generalizations of Directly Similar Theorem?

There is an attractive theorem that says that if two plane figures
are directly similar, then so is any convex combination of them.
Below, $P_1$ and $P_2$ are directly similar polygons: they have
the ...

**2**

votes

**1**answer

169 views

### Model of hyperbolic geometry with finite number of parallel line

Does there exist a model of hyperbolic geometry such that only finite number of distinct parallel lines through a point which does not intersect given line?
Edit (Misha): I usually do not edit other ...

**12**

votes

**1**answer

285 views

### Lower-Hölder embeddings of the sphere

My question is very simple:
Given $d\ge 3$, does there exist $s\in (0,1)$ and an embedding $f:S^{d-1}\to \mathbb{R}^d$ such that
$$
|f(x)-f(y)| \ge |x-y|^s \quad\textrm{if } |x-y|<r,
$$
for ...

**14**

votes

**0**answers

422 views

### An open problem in convex geometry

Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist? ...

**9**

votes

**0**answers

157 views

### Multiplicity of ball covering

Background. My questions are motivated by the following:
A. Conway and Sloane in "On the covering multiplicity of lattices" (Discrete
and Computational Geometry, 8 (1992) 109-130) considered the ...

**12**

votes

**2**answers

557 views

### Right triangle with edge lengths equal to regular unit polygon edge lengths

This question came up naturally recently from a blog post of John Baez. There is an observation of Euclid that edges of a pentagon, hexagon, and decagon inscribed in a unit circle form the edges of a ...

**3**

votes

**1**answer

140 views

### Restricted isometry

Here is a problem that has been bugging me for a while.
Let $\| \|$ be a norm over $\mathbb{R}^n$, let $C$ be a convex subset of $\mathbb{R}^n$ with non-empty interior, and let $f: (C,\|\|) ...

**1**

vote

**0**answers

89 views

### Entangled helical knots

Consider a pair of disjoint, congruent helices $H_1$ and $H_2$
passing through one another in the following sense.
(Caveat lector: This question is not of general interest! It is also long.)
$H_1$ is ...

**11**

votes

**1**answer

383 views

### Heronian triangle with two sides that are prime

Can any prime number form a Heronian triangle with a second prime as another side? I cannot find a second prime to form a Heronian triangle with either 23 or 167. I have checked up to the 10^7th prime ...

**1**

vote

**1**answer

162 views

### Probabilistic Johnson-Lindenstrauss Lemma for arbitrary points

Consider the following standard formulation of the Johnson-Lindenstrauss lemma:
Lemma (JL).
For any $0<\epsilon < 1$ and any integer $n$, let $k$ be a positive integer such that $k\geq ...

**6**

votes

**0**answers

166 views

### Reversing shortest paths among unit disks

Twas the night before Christmas, and throughout M.O.
Not a question was posted, not even by Joe.
Well, let me remedy that. :-)
Let the plane contain a number of ...

**3**

votes

**2**answers

184 views

### Equipartition of the circle [closed]

Browsing an old technical studies pupil's school book, I have found the description of a method to place at equal distance $N$ points on the circumference of a circle. I am looking for a proof of this ...

**3**

votes

**1**answer

160 views

### Closed surface with uncountably many conic points?

Let $M^2$ be a closed surface, say the 2-sphere. Is there any example of metric on it such that there are uncountably many points are conic and the metric is smooth elsewhere?
We call $p\in M$ a ...

**-2**

votes

**1**answer

157 views

### Equivalence relations on powerset of R^2

Let A and B be two subsets of R^2. I define the relation T(A,B) to hold between A and B iff there exists a translation f on R^2 such that the image set of A under f is B. It is easy to prove that T is ...

**5**

votes

**1**answer

153 views

### Shortest curve with given convex hull

Suppose $S\subset\mathbb{R}^2$ is compact and convex. Suppose $\Gamma:[0,1]\to S$ is a continuous curve that passes through every extreme point of $S$, i.e., the convex hull of $\Gamma([0,1])$ is $S$. ...

**3**

votes

**0**answers

88 views

### Lattices achieving best density

Let $\Lambda \subset \mathbb{R}^n$ be an Euclidean lattice with generator matrix $B$. Define the center density $\delta(\Lambda)$ in the usual way as $\delta(\Lambda) = \rho^n/|\det{B}|$, where $\rho$ ...

**18**

votes

**0**answers

323 views

### The Eyeball Theorem generalized

I have not seen the 2D Eyeball Theorem—that tangents from the centers of two circles, each encompassing the other, intersect each circle in the same segment length—generalized to higher ...

**10**

votes

**1**answer

633 views

### What is the longest algebraic curve?

Consider a convex body $\Omega\subset \mathbb{R}^2$. Let $L(d)$ be the maximum over all curves $C$ of degree $d$ of the length of $C\cap\Omega$.
Is $L(d)\leq d P(E)/2$, where $P(E)$ is the ...

**0**

votes

**1**answer

129 views

### Lipschitz boundary vs rectifiable curve boundary

I was looking at an old paper about domains with Lipschitz boundary. I am wondering, suppose that the boundary of a compact domain homeomorphic to a disk is a rectifiable injective curve : is this ...

**13**

votes

**3**answers

647 views

### Are infinite planar graphs still 4-colorable?

Imagine you have a finite number of "sites" $S$ in the positive quadrant
of the integer lattice $\mathbb{Z}^2$,
and from each site $s \in S$, one connects $s$ to every lattice point to which it
has a ...

**2**

votes

**2**answers

108 views

### Nonplanar equilateral lattice “pentagons”

It is well-known that no two-dimensional point lattice contains a regular pentagon. (See for example http://mathworld.wolfram.com/LatticePolygon.html.) The same is true for lattices in $\mathbb{R}^n$, ...

**34**

votes

**3**answers

1k views

### What fraction of the integer lattice can be seen from the origin?

Consider the integer lattice points in the positive quadrant $Q$ of $\mathbb{Z}^2$.
Say that a point $(x,y)$ of $Q$ is visible from the origin if the
segment from $(0,0)$ to $(x,y) \in Q$ passes ...

**3**

votes

**2**answers

193 views

### Place N points in a 3d cube in a way that maximizes the minimum of their pairwise distances

Place $N$ points in a 3d cube in a way that maximizes the minimum of their pairwise distances.
The problem can easily be solved for $N\lt5$, but how to proceed for larger $N$?

**5**

votes

**1**answer

174 views

### Which lenses can be squared?

A lune and a lens are both planar figures delineated by two circular arcs; the difference is that a lune has a concave and a convex arc (it is one circle minus another) whereas a lens has two convex ...

**4**

votes

**1**answer

140 views

### Bounding the perimeter of a geodesic triangle in spaces of non-positive curvature

This is probably an easy question, but I don't know any Riemannian geometry and a literature search hasn't helped. Any help (e.g. providing a reference) would be greatly appreciated.
For a triangle ...

**1**

vote

**1**answer

33 views

### Length of the transversal for surfaces with cusps

In Peter Buser's Geometry and Spectra of Compact Riemann Surfaces he shows that the length of the transverse curve to a geodesic in a pants decomposition on a compact hyperbolic surface has length a ...

**3**

votes

**1**answer

192 views

### Map from a convex polygon that increases distance

At the risk of asking an extremely stupid question, suppose that $P\subset\mathbb{R}^2$ is a convex polygon with area $1$ that contains the origin, and let $r$ denote the farthest distance between the ...

**4**

votes

**3**answers

264 views

### On discrete version of curve shortening flow

One can define an analogous version of the curve shortening flow for polygons in $\mathbb R^2$, namely defined by the differential equation $\dot{p_i}(t)=\frac{v_i(t)}{|v_i(t)|^2}$, where $p_i$ is the ...

**16**

votes

**2**answers

398 views

### What kind of probability distribution maximizes the average distance between two points?

If $f$ is a probability distribution on the unit disk in $\mathbb{R}^2$, and $X_1$ and $X_2$ are two independent samples from $f$, then what is the distribution $f^*$ that maximizes the average ...

**9**

votes

**0**answers

215 views

### Periodic orbits of a spinning ball in a square

Periodic orbits of a billiard ball bouncing in a square have been well-studied.
I am seeking similar analysis of what is sometimes called a rough ball, one
whose high friction causes it to pick up ...

**6**

votes

**1**answer

101 views

### Maximizing ratio volume/diameter^n by an affinity

Suppose we have a convex compact body $D\subset \mathbb R^n$. We can try to apply affine transformation keeping the volume and decreasing the diameter of $D$.
It is clear that there is a constant ...