Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
Questions tagged [mg.metric-geometry]
3939 questions
86
votes
4 answers
Is the sphere the only surface with circular projections? Or: Can we deduce a spherical Earth by observing that its shadows on the Moon are circular?
Several ancient arguments suggest a curved Earth, such as
the observation that ships disappear mast-last over the
horizon, and
Eratosthenes'
surprisingly accurate calculation of the size of the
Earth
by measuring a difference in shadow length…
Joel David Hamkins
- 224,022
40
votes
3 answers
Distributing points evenly on a sphere
I am looking for an algorithm to put $n$-points on a sphere, so that the minimum distance between any two points is as large as possible.
I have found some related questions on stackoverflow but those algorithms are not an exact solution more random…
CPJ
- 732
39
votes
1 answer
What shapes can be gears?
I am interested in gears. Sadly most of the writing on this is very practical and does not get into abstract theory. I have been trying to formalize these ideas to be able to ask what shapes can function as gears and at what ratios. I have had two…
Spencer Woolfson
- 481
32
votes
2 answers
Gromov-Hausdorff distance between a disk and a circle
The Hausdorff distance between the closed unit disk $D^2$ of $\mathbb R^2$ (equipped with the standard Euclidean distance) and its boundary circle $S^1$ is obviously one.
Interestingly, the Gromov-Hausdorff distance between $D^2$ and $S^1$ is…
rozu
- 803
31
votes
1 answer
Can R^3 be expressed as a disjoint union of pairwise linked circles?
We can express $\mathbb{R}^3$ as a disjoint union of circles. There are some constructive ways of doing this, although it's easier to construct them sequentially by transfinite induction, applying the following step to each point $P_{\alpha}$ in the…
Adam P. Goucher
- 12,211
30
votes
3 answers
Diameter of m-fold cover
Let $M$ be a closed Riemannian manifold.
Assume $\tilde M$ is a connected Riemannian $m$-fold cover of $M$.
Is it true that
$$\mathop{diam}\tilde M\le m\cdot \mathop{diam} M\ ?\ \ \ \ \ \ \ (*)$$
Comments:
This is a modification of a problem of A.…
Anton Petrunin
- 43,739
29
votes
2 answers
Maneuvering with limited moves on $S^2$
This question comes to me via a friend, and apparently has something to do with quantum physics. However, stripped of all physics, it seems interesting enough on its own. I assume someone has asked this question before, but I have no idea what to…
Eric Tressler
- 1,298
23
votes
2 answers
Is there a neat formula for the volume of a tetrahedron on $S^3$?
There is a nice formula for the area of a triangle on the 2-dimensional sphere;
If the triangle is the intersection of three half spheres, and has angles $\alpha$, $\beta$ and $\gamma$, and we normalize the area of the whole sphere to be $4\pi$ then…
Ehud Friedgut
- 435
23
votes
4 answers
Intrinsic metric with no geodesics
It seems that I have the needed example, but I want it to be simple and self-explaining...
Construct a nontrivial complete metric space $X$ with intrinsic metric which has no nontrivial minimizing geodesics.
Definitions:
A metric $d$ is called…
Anton Petrunin
- 43,739
20
votes
3 answers
Gromov-Hausdorff distance between p-adic integers.
What is the distance in the sense of Gromov-Hausdorff between $\mathbb{Z}_{p_1}$ and $\mathbb{Z}_{p_2}$ with the usual p-adic metrics?
I got stuck and simply have no idea how to deal with such questions: I've got two metric trees and have to observe…
Dmitrii Korshunov
- 2,147
20
votes
5 answers
Advanced view of the napkin ring problem?
The "napkin-ring problem" sometimes shows up in 2nd-year calculus courses, but it can fit quite neatly into a high-school geometry course via Cavalieri's principle.
However, the conclusion remains astonishing. Is there some advanced viewpoint from…
Michael Hardy
- 11,922
- 11
- 81
- 119
20
votes
7 answers
Is there always a maximum anti-rectangle with a corner square?
Let $C$ be an axis-parallel orthogonal polygon with a finite number of sides. Define an anti-rectangle in $C$ as a set of small squares in $C$, such that no two of them are covered by a single large rectangle in $C$. Define a maximum anti-rectangle…
Erel Segal-Halevi
- 3,585
18
votes
3 answers
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 the infimum of the real $k$ such that there is a…
Joël
- 25,755
17
votes
1 answer
Is a facet always a maximal area section of a simplex?
Let $T\subset \mathbb{R}^n$ be a fixed simplex, $H\subset \mathbb{R}^n$ be a variable affine hyperplane. Is it true that the maximal area (i.e. the $(n-1)$-dimensional volume) of $T\cap H$ is attained when $H$ contains a facet of $T$?
Fedor Petrov
- 102,548
16
votes
3 answers
Towards a metric characterization of Euclidean spaces
I want to obtain a metric characterization of the classical finite dimensional spaces of Euclidean geometry.
Motivation: Suppose $A$ and $B$ live in an $n$-dimensional Euclidean space. They are each assigned the task of constructing an equilateral…
Marcos Cossarini
- 1,917