Euclidean, hyperbolic, discrete, convex, coarse geometry, comparisons in Riemannian geometry, symmetric spaces.
7
votes
2answers
130 views
How to find a tetrahedron that covers four points?
I’m looking for an explicit formula for the vertices of a regular tetrahedron that covers four given points. In particular:
Given four distinct real numbers $a_1$, $a_2$, $a_3$, $a_4$, is there a ...
9
votes
3answers
410 views
Does the centroid depend continuously on the curve?
Let $\gamma$ be a piecewise smooth curve in $\mathbb{R}^n$. Recall that the centroid of $\gamma$ is the point $(\overline{x}, \overline{y})$ where $\overline{x}$ is the average value of $x$ on ...
11
votes
2answers
229 views
Triangle with largest perimeter in a convex region
What is the largest value of $r$ such that the following statement is always true?
"Let $C$ be a convex region with area $1$. There must exist a triangle contained in $C$ whose perimeter is at least ...
6
votes
1answer
129 views
quantitative version of the rigidity of the 2-sphere
I am looking for a quantitaive version of the following theorem:
A compact surface with $K\equiv 1$ is isometric to the round sphere.
Of course I get the Berger, Brendle-Schoen Theorem which insures ...
3
votes
1answer
170 views
A question on differential forms and integral invariants
The following question comes up in the study of metrics with the same unparameterized geodesics in Riemannian and Finsler geometry:
Question. Let $M$ be a closed manifold of dimension $2n+1$ and let ...
8
votes
4answers
265 views
More than $n$ approximately orthonormal vectors in $R^n$
This question was asked at math.stackexchange, where it got several upvotes but no answers.
It is impossible to find $n+1$ mutually orthonormal vectors in $R^n$.
However, it is well established ...
0
votes
0answers
25 views
Fast calculation of the area of intersection between a sphere and a cylinder [migrated]
In my current research, I am looking at calculating the local porosity of a porous media in cylindrical coordinate (notably, two co-centric cylinders).
To obtain an accurate approximation, I need to ...
4
votes
2answers
114 views
Bound on Minimal Length of Vectors in Lattice and its Dual Lattice
Let $\Lambda$ be a lattice in $\mathbb{R}^n$ and $\Lambda^\ast$ its dual lattice. Let $d=\min_{v\in\Lambda} (v,v)$ and $d^\ast =\min_{v\in\Lambda^\ast} (v,v)$ be the minimal squared lengths of vectors ...
3
votes
0answers
50 views
3D objects with projections of constant area
Objects of constant width are well-known, and I have in my hand a (non-spherical) solid of constant diameter. The question arises:
Are there any non-spherical 3-dimensional objects such that their ...
4
votes
1answer
160 views
Boundaries of relatively hyperbolic groups
When the interior of an n-manifold $M$ has a pinched negative curvature metric of finite volume, then its fundamental group $\Gamma=\pi_1M$ is relatively hyperbolic relative to the parabolic groups ...
7
votes
0answers
191 views
Surfaces with many (but not solely) closed geodesics?
Let $S$ be a closed surface embedded in $\mathbb{R}^3$,
let's say of genus zero.
I seek examples of $S$ with the following property:
If one selects a random any point $p$ on $S$, and a random
...
9
votes
0answers
144 views
Unit ball of smallest volume in a Hilbert geometry
In a letter to Felix Klein published in Mathematische Annalen 1895 (see here), Hilbert generalized the Cayley-Klein model of hyperbolic geometry by defining a metric on the interior of a convex body ...
0
votes
1answer
72 views
Constructing measures with support in a given set
I've recently come across the Frostmann Lemma (http://en.wikipedia.org/wiki/Frostman_lemma). Its proof involves constructing a measure with certain properties on a given subset of $\mathbb{R}^n$ (I'm ...
1
vote
2answers
62 views
Extreme points and centroid
It is well-known that the centroid of a triangle is the intersection point of its three medians. The medians happen to be area bisectors, but it seems that most (all?) other lines through the ...
0
votes
1answer
70 views
harmonic maps from cone to $S^2$ locally lipschitz?
Are the harmonic maps from a 2-dimensional cone to $S^2$ locally lipschitz or Holder continuous?
1
vote
2answers
54 views
Sampling uniformly from all possible line segments of a given length that fit inside a container
Consider that task of randomly placing a line segment of some length $L$ near a plane s.t. a point $p$ at the center of the line segment is at most a distance $H$ from the plane and intersections ...
12
votes
3answers
348 views
Characterization of discs
Let $D$ be a bounded simply connected region (open subset homeomorphic to the disc)
in the plane, containing the origin.
Suppose that for every line $L$ through the origin the intersection ...
5
votes
1answer
115 views
Maximal regions with given diameter
Let us call a bounded region $D$ in the plane maximal if the conditions $D\subset D'$ and
$\mathrm{diam} D'=\mathrm{diam} D$ imply $D'=D$.
Is it possible to describe all maximal regions?
The only ...
2
votes
1answer
34 views
Seeking criteria for “threadable” pairs of centrosymmetric polyhedra
Let $A$ and $B$ be origin-centered centrosymmetric polyhedra in $\mathbb{R}^3$:
"for every point $(x, y, z)$ [...] there is an indistinguishable point $(-x, -y, -z)$."
Say that $A$ and $B$ are ...
2
votes
0answers
36 views
Measure of points with small neighborhood in convex bodies
Let $K \subseteq \mathbb{R}^n$ be a "fat" convex body, i.e. one that contains a ball of radius 1. I'm interested in the following question about points $y \in K$: If you take a normally distributed $e ...
0
votes
0answers
59 views
Problem on infinite dimensional metric space, with rigidity assumption
By inspiring from this answer of S. Ivanov, here is a specialization with a rigidity assumption.
Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces verifying :
...
4
votes
1answer
210 views
Probability distribution or the distance between two points in $n$-dimensional Euclidean space after a random perturbation of one point
Take two points, $p_0$ and $p_k$, in $n$-dimensional Euclidean space, where $d(p_0,p_k)$ is the distance between the points. Now, draw an $n$-sphere of radius $r$ centered on $p_0$ and uniformly ...
11
votes
2answers
307 views
The intersection of a circle and a rank 3 subgroup of the plane
Let $A$ be a rank 3 subgroup of the Euclidean plane, i.e. $A = \mathbb{Z} v_1 + \mathbb{Z} v_2 + \mathbb{Z} v_3$, where $v_1, v_2, v_3 \in \mathbb{R}^2$ are three $\mathbb{Q}$-linearly independent ...
7
votes
1answer
396 views
Is this knot invariant already treated somewhere in the literature?
Fix a knot type $K \subset S^3$, and consider the set $$Y_K = \{ \mbox{Diagrams of }K \} / \mbox{planar isotopy}.$$
We can turn $Y_K$ into a metric space by considering the distance induced by ...
1
vote
0answers
46 views
Twisted calibrations and sufficient conditions on homology of sub-manifolds
I think my question is somehow easy to solve, but I'm not very familiar with algebraic topology, so I'm not able to figure out the solution for myself. I'm working on a problem in metric geometry and ...
4
votes
1answer
165 views
Computational approach deciding whether a set of Wang Tile could tile the space up to some size
As an applied person, I'm facing one practical problem deciding whether a set of Wang tile could tile the plane periodically or aperiodically. Although both problems seem undecidable, but I'm on a ...
5
votes
3answers
223 views
How hard is it to determine if a weighted graph can be isometrically embedded in R^3?
Consider a graph $G$ with nonnegative edge weights.
Question: In $\mathbb{R}^3$, how hard is it to assign coordinates to vertices such that the Euclidean length of each edge is equal to its weight?
...
9
votes
1answer
252 views
A question about tiling Hilbert Space
Let H be an infinite dimensional and separable Hilbert Space. Let e be a positive real number-which can be arbitrarily small. Does there exist a denumerably infinite set S of pairwise disjoint and ...
3
votes
1answer
106 views
Planar linkage that traces a circle from its exterior?
Q.
Is there a linkage in the plane that traces out a circle $C$
in such a manner that the interior of the disk bounded
by $C$ is never intersected by any link througout the motion?
What I ...
1
vote
0answers
101 views
Why is proving $C^{\infty}$ regularity of sub Riemannian geodesics so hard?
In Montgomery's A Tour of Subriemannian Geometries, Their Geodesics and Applications, problem 10.1 in Chapter 10 asks "Is every minimizing geodesic smooth ?".
Can someone explain what are the major ...
11
votes
0answers
322 views
Blocking light with mirrored convex objects
There is a long-unsolved problem posed by Janos Pach,
sometimes known as the enchanted forest problem,
which asks if it is possible to block a point light source
in the plane
from reaching
infinity by ...
1
vote
2answers
125 views
Local vs distance function metric structures
The geodesic distance $d(p_1, p_2)$ on a geodesically complete, Riemanian manifold is a metric in the sense of a metric space metric. I'd like to know when other infinitesimal metric structures (e.g. ...
2
votes
0answers
61 views
Approximation of a convex body by a contained polytope
This question deals with approximating a convex body (a compact convex set of $\mathbb{R}^d$ with non-empty interior) by convex polytopes.
For a given $\delta$, let $n_\delta$ be the number of faces ...
1
vote
2answers
165 views
Möbius transformation by 3 points in the Minkowski model
Goal
I'm interested in describing Möbius transformations in the plane, and I'd like to define them in terms of three points and their images.
What I have tried
I know that a projective ...
9
votes
0answers
141 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
281 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 ...
12
votes
4answers
701 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
93 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
249 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
49 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
127 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
74 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 ...
0
votes
2answers
328 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
60 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
104 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 ...
20
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
165 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 ...
2
votes
0answers
71 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 ...
5
votes
2answers
231 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
168 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?
...
