Euclidean, hyperbolic, discrete, convex, coarse geometry, comparisons in Riemannian geometry, symmetric spaces.

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9
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0answers
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
2answers
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
4answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
2answers
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
0answers
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
1answer
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
3answers
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
1answer
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
1answer
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
2answers
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
1answer
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
5answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
2answers
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
2answers
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
3answers
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
2answers
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
3answers
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
2answers
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
1answer
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
1answer
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
1answer
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
1answer
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
3answers
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
2answers
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
0answers
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
1answer
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 ...