All Questions
4,828 questions
14
votes
3
answers
3k
views
What are the tricks for computing/estimating Gromov-Hausdorff distance?
Gromov-Hausdorff distance (Wikipedia) between two compact manifolds measures how far away the manifolds are from being isometric.
In many cases it is possible to do coarse estimates and conclude that ...
- 1,414
14
votes
2
answers
663
views
Are there good mutually interpretable axioms for synthetic Euclidean and hyperbolic geometry?
There are familiar analytic equiconsistency proofs for Euclidean and hyperbolic geometry. Those proofs are so robustly geometric that it seems like they must have synthetic analogues.
Looking into ...
- 11.1k
14
votes
2
answers
878
views
What is the prime spectrum of a Cauchy series ring?
Let $k$ be a field, and let $| \ |$ be a norm on $k$. The norm induces a metric. To construct the completion $\hat{k}$ as a normed field, the standard recipe is to take the quotient of the ring $\...
- 65.4k
14
votes
2
answers
319
views
Shortest path through $n^{1/3}$ points out of $n$
Say I sample $n$ points uniformly at random in the unit cube in $\mathbb{R}^3$, and then I look for the shortest path through $n^{1/3}$ of those points (rounding up, say). What happens to the length ...
- 141
14
votes
1
answer
886
views
Distance to an apartment of the affine building of GL(N)
Here $F$ is a locally compact non-archimedean non-discrete field.
Let $X$ be the reduced (affine) Bruhat-Tits building of ${\rm GL}(n,F)$. Fix a maximal split torus $T$. Let $B$ be a Borel subgroup ...
- 6,349
14
votes
1
answer
280
views
How many distances are required to calculate all distances among $n$ points in the Euclidean plane?
I want to know all the pairwise distances between points $P_1,P_2,\ldots,P_n$ in the Euclidean plane (or equivalently, I want to reconstruct the set of points up to congruence). Let's say I have an ...
- 523
14
votes
1
answer
453
views
Does existence of midpoints imply intrinsic?
It is well-known, that a complete metric space, where any two points have a midpoints ($\forall x,y~ \exists z:~d(x,z)=d(y,z)=\frac{d(x,y)}{2}$) is strictly intrinsic, in the sense that any $x,y$ can ...
- 5,529
14
votes
3
answers
2k
views
Optimal wireframe sphere
Suppose you have a length $L$ of metal pipe at your disposal,
and you would like to build a wireframe unit-radius sphere,
by bending, cutting, and welding the pipe into a connected structure $F$.
Your ...
- 151k
14
votes
1
answer
922
views
What are the applications of the Mazur-Ulam Theorem?
Every bijective isometry between normed spaces is affine. This well-known and beautiful statement, the Mazur-Ulam Theorem, was proved in 1932, but the proof has been simplified and polished in years, ...
- 60.5k
14
votes
2
answers
2k
views
The Disco Ball Problem
Let me first give some of a background as to where I got this problem. I had a math teacher ask me a few months ago: "How many 1 unit by 1 unit squares could one fit on a sphere with a radius of 32 ...
- 141
14
votes
2
answers
2k
views
Surface area of an $\ell_p$ unit ball?
Are there any known formulas or approximations for the surface area of a unit ball in $d$ dimensions under the $\ell_p$ norm? As obvious examples, it is of course well-known that the surface area of ...
- 141
14
votes
1
answer
837
views
Applications of the GCD metric
In the pre-MO era, I once realized that on the integers, the function
$$
d(m, n) := \sqrt{\log \frac{\sqrt{mn}} {\text{gcd}(m,n)}}\ ,
$$
is a metric (all properties are easily verified; in fact ...
- 28.6k
14
votes
1
answer
642
views
Which convex bodies can be captured in a knot?
Which convex bodies can be captured in a knot?
This question is based on the discussion in "Is it possible to capture a sphere in a knot?".
We assume that the knot is made from an ...
- 45k
14
votes
1
answer
1k
views
Egg-ovoid rolling down an inclined plane
I am seeking a mathematical analysis of an egg-ovoid rolling down an inclined plane,
for pedagogical reasons.
It is well-known folk lore that the shape of an egg prevents it from rolling away from
...
- 151k
14
votes
2
answers
1k
views
Polygonal billards programs
I'm looking for software that will give billiard trajectories in arbitrary plane polygons. After much work I was able to produce this figure.
(source)
It was a good exercise, but at this point I ...
- 22.8k
14
votes
1
answer
2k
views
Dao's theorem on six circumcenters associated with a cyclic hexagon
This questions from Ngo Quang Duong's paper
In 2013, O. T. Dao published without proof a theorem with title Another seven circles theorem in Cut the Knot, a free site for popular expositionsof many ...
- 1,044
14
votes
0
answers
270
views
Regular $n$-gon with diagonals: bounds on area of largest cell?
Consider a regular $n$-gon of side length $1$ with diagonals. Here is an example with $n=11$ (from geogebra applet).
I've been trying to find, in terms of $n$, bounds on the area of the largest cell, ...
- 3,527
14
votes
0
answers
205
views
Have there been further developments on this scheme for polytope approximations to the unit ball of $\ell_p^n$?
A long time ago I happened to look at, and save (on a floppy disk!) for future reading, a copy of the following article:
W. T. Gowers, Polytope approximations of the unit ball of $l^n_p$.
In Convex ...
- 25.8k
14
votes
0
answers
479
views
Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between edges are rational multiples of $\pi$?
After reading these very interesting questions, I came up with another one:
Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between all pairs of edges meeting ...
- 532
14
votes
0
answers
552
views
Who conjectured that a transitive projective plane is Desarguesian?
The only known finite projective plane with a transitive automorphism group is the Desarguesian plane $PG(2,q)$ and it seems likely that there are no others, although this is not (quite) proved.
...
- 12.7k
14
votes
0
answers
4k
views
Minimum tiling of a rectangle by squares
Given the $n\times m$ rectangle, I want to compute the minimum number of integer-sided squares needed to tile it (possibly of different sizes).
Is there an efficient way to calculate this?
- 1,015
13
votes
8
answers
3k
views
Applications of the notion of of Gromov-Hausdorff distance
I am looking for applications of the notion of Gromov-Hausdorff convergence to prove theorems that a priori have nothing to do with it. Examples that I am aware of (thanks to wikipedia and google):
...
13
votes
2
answers
917
views
Can the circle be characterized by the following property?
In the Euclidean plane, is the circle the only simple closed curve that has an axis of symmetry in every
direction?
- 6,215
13
votes
3
answers
3k
views
Are uniformly continuous functions dense in all continuous functions?
Suppose that $X$ is a metric space. Is the family of all real-valued uniformly continuous functions on $X$ dense in the space of all continuous functions with respect to the topology of uniform ...
- 233
13
votes
5
answers
1k
views
Packing obtuse vectors in $\mathbb{R}^d$
I came across this attractive theorem:
Theorem. In $\mathbb{R}^d$, there can be at most $d+1$ vectors that
form an obtuse angle with one another.
This was proved1 as a corollary of a lemma about ...
- 151k
13
votes
4
answers
1k
views
Can Lipschitz maps increase the Lebesgue dimension ?
Given a map $f:X\rightarrow Y$ of compact metric spaces, such that there is a $C\in \mathbb{R}$ with $d(f(x),f(x'))\le C\cdot d(x,x')$.
Does this already imply, that the Lebesgue dimension of $f(X)$ ...
- 11.1k
13
votes
7
answers
2k
views
Upper bound on the area of a midpoint pentagon?
Starting with a convex pentagon P, we define the "middle polygon" Q, whose vertices are the middle points of the sides of the initial pentagon. The ratio between the areas of this polygons seem to ...
- 566
13
votes
3
answers
835
views
What fraction of n-point sets in the unit ball have diameter smaller than 1?
This question is inspired by a recent talk by Matt Kahle on random geometric complexes.
Some simple notation: let $\mathcal{B} \subset \mathbb{R}^d$ be the unit ball in $d$-dimensional Euclidean ...
- 15.5k
13
votes
2
answers
2k
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 ...
- 133
13
votes
5
answers
2k
views
Is there a complete classification of constant mean curvature surfaces?
I'm no expert in this field, but I am familiar with the classification of rotationally symmetric surfaces with constant mean curvature by Delaunay. I am aware that once we drop embeddedness and ...
- 325
13
votes
3
answers
459
views
A comparison question for non-positively curved disks
Let $A$ and $B$ be two closed, 2-dimensional, non-positively-curved Riemannian disks (not necessarily with convex boundary). Suppose that their boundaries $\partial A$ and $\partial B$ have the same ...
- 56.6k
13
votes
2
answers
473
views
Can any simplex shadow-project to a regular simplex?
Every triangle $A$ can be oriented in $\mathbb{R}^3$
so that its orthogonal projection (shadow) onto the $xy$-plane is an
equilateral triangle $Q$:
&...
- 151k
13
votes
2
answers
865
views
For what metrics are circles solutions of the isoperimetric problem?
A classical result is that solutions of the isoperimetric problem on the plane, the sphere, and the hyperbolic plane are circles. Are there any other Riemannian metrics on these spaces that share this ...
- 13.5k
13
votes
3
answers
1k
views
Efficient visibility blockers in Pólya's orchard problem
Pólya's orchard problem asks for which radius $\rho$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the exterior of the orchard.
It has been ...
- 151k
13
votes
3
answers
2k
views
Metric angles in Riemannian manifolds of low regularity
Given three points $a,b,c$ in a (geodesic) metric space $X$, one defines a comparison angle $\angle(a,b,c)$ by the cosine law:
$$
\angle(a,b,c) = \arccos \frac{|ab|^2 + |ac|^2 - |bc|^2}{2\cdot|ab|\...
- 32.4k
13
votes
2
answers
3k
views
How many squares can be formed by using n points?
How many squares can be formed by using n points on a 3 dimensional space?
Like using 4 points, there is 1 square be formed
Using 5 points, still 1 square
Using 6 points, 3 squares can be formed
- 241
13
votes
2
answers
664
views
Helix translates as geodesics
I believe one can fill $\mathbb{R}^3$ with
horizontal translates of the helix
$(\cos t, \sin t, t) \;,\; t \in \mathbb{R}$,
so that every point of $\mathbb{R}^3$
lies in exactly one helix.
I am ...
- 151k
13
votes
2
answers
2k
views
Is it a new discovery on conic section?
I discovered a problem in plane geometry (there are some nice special cases) as follows:
Let $ABC$ be a triangle and $\Omega$ be arbitrary circumconic. Let two points $A_b, A_c \in BC$, $B_c, B_a \in ...
- 1,776
13
votes
2
answers
909
views
A problem on convex geometry
Consider a convex body $K \subset \mathbb{R}^n$ containing the origin in its interior. Although the body is not necessarily symmetric, let us say that two points in its boundary $\partial K$ are ...
- 13.5k
13
votes
2
answers
664
views
Complexity of a weirdo two-dimensional sorting problem
Please forgive me if this is easy for some reason.
Suppose given $S$, a set of $n^2$ points in $\mathbb{R}^2$.
I want to choose a bijective map $f$ from $S$ to the set of lattice points in $\lbrace ...
- 19.2k
13
votes
2
answers
514
views
Can you see smoothness of the boundary of a convex body from its shadow?
Let $d \geq 3$ and suppose that $K \subset \mathbb{R}^{d}$ is a convex body (compact, convex, non-empty interior). Is the following true?
The boundary $\partial K$ is a $C^1$-manifold if and only ...
- 491
13
votes
3
answers
421
views
Maximal distance between $2d+1$ points on the $(d-1)$-sphere
If one arranges $2d$ points on the sphere $\mathbf S^{d-1}\subset\Bbb R^d$ at the vertices of the crosspolytope, then one can achieve a minimal spherical distance of $\pi/2$ between any two points, ...
- 13.6k
13
votes
1
answer
730
views
Illustrating that universal optimality is stronger than sphere packing
I'm a physicist interested in the conformal bootstrap, one version of which was recently shown to have many similarities to the problem of sphere packing. Sphere packing in $\mathbf{R}^d$ has been ...
- 242
13
votes
2
answers
807
views
Prehistory of Gromov-hyperbolic spaces/groups
When speaking about hyperbolic groups/spaces, one usually refers to Gromov's monograph Hyperbolic groups for their introduction. However, coarse notions of hyperbolicity can be found in some of his ...
- 8,401
13
votes
3
answers
650
views
General principles which lead to good questions in many concrete situations [closed]
I believe that in various fields of mathematics there are general principles which might lead to good questions and good results in many concrete situations. I would like to have a list of such ...
13
votes
2
answers
918
views
Acute triangulation
Assume that $S$ is a finite 2-dimensional simplicial complex equipped with a metric $d$
such that each triangle is isometric to a plane triangle (so $(S,d)$ is a polyhedral space).
Is it possible ...
- 45k
13
votes
2
answers
2k
views
Altitudes of a triangle
The three altitudes of a triangle are concurrent -- this is true in all three constant curvature geometries (Euclidean, hyperbolic, spherical), but, as far as I know, the proofs are different in the ...
- 96.4k
13
votes
2
answers
1k
views
Average degree of contact graph for balls in a box
Imagine you dump congruent, hard, frictionless balls in a box,
letting gravity compress the balls into a stable configuration
(I believe such configurations are called
jammed.)
Assume the box ...
- 151k
13
votes
1
answer
844
views
Euclidean tangent cone implies Riemannian manifold
It is known that given a Riemannian manifold, then the tangent cone (as a metric space) at any point $p$ is isometric to the tangent space at $p$, with the metric given by the metric tensor.
Is ...
- 2,129
13
votes
4
answers
1k
views
When sticks fall, will they weave?
Imagine $n$ $z$-vertical sticks uniformly spaced around a unit-radius circle in the $xy$-plane.
At $t{=}0$, each is randomly $\epsilon$-perturbed from the vertical, and they fall under
the influence ...
- 151k