All Questions
1,585 questions
73
votes
10
answers
11k
views
Riemannian surfaces with an explicit distance function?
I'm looking for explicit examples of Riemannian surfaces (two-dimensional Riemannian manifolds $(M,g)$) for which the distance function d(x,y) can be given explicitly in terms of local coordinates of ...
- 114k
20
votes
2
answers
7k
views
Question about functional derivatives
This page on Wikipedia defines the so-called functional derivative as follows: "Given a manifold $M$ representing (continuous/smooth) functions $\rho$ (with certain boundary conditions, etc.) and a ...
- 3,515
25
votes
3
answers
13k
views
Fourier transform of the unit sphere
The Fourier transform of the volume form of the (n-1)-sphere in $\mathbf R^n$ is given by the well-known formula
$$
\int_{S^{n-1}}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u) = (2\pi)^{\nu ...
- 31.5k
41
votes
4
answers
16k
views
Product of Borel sigma algebras
If $X$ and $Y$ are separable metric spaces, then the Borel $\sigma$-algebra $B(X \times Y)$ of the product is the $\sigma$-algebra generated by $B(X)\times B(Y)$. I am embarrassed to admit that I ...
- 31.5k
28
votes
8
answers
6k
views
Representability of finite metric spaces
There have been a couple questions recently regarding metric spaces, which got me thinking a bit about representation theorems for finite metric spaces.
Suppose $X$ is a set equipped with a metric $d$...
- 4,014
49
votes
0
answers
3k
views
Concerning proofs from the axiom of choice that ℝ³ admits surprising geometrical decompositions: Can we prove there is no Borel decomposition?
This question follows up on a comment I made on Joseph O'Rourke's
recent question, one of several questions here on mathoverflow
concerning surprising geometric partitions of space using the axiom
of ...
- 236k
41
votes
3
answers
15k
views
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 ...
- 742
5
votes
1
answer
630
views
Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution
Examples of infinite dimensional involutions
Edit 2/25/23, as suggested by YCOR below: (Start)
The first return on a Google search on involution--from late Latin 'a rolling up'--gives the Oxford ...
- 10.5k
88
votes
4
answers
11k
views
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
...
- 236k
70
votes
4
answers
11k
views
$C^1$ isometric embedding of flat torus into $\mathbb{R}^3$
I read (in a paper by Emil Saucan) that the flat torus may be isometrically embedded
in $\mathbb{R}^3$ with a $C^1$ map by the Kuiper extension of the Nash Embedding Theorem,
a claim repeated in this ...
- 151k
49
votes
4
answers
4k
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 ...
- 151k
45
votes
7
answers
9k
views
What's an example of a space that needs the Hahn-Banach Theorem?
The Hahn-Banach theorem is rightly seen as one of the Big Theorems in functional analysis. Indeed, it can be said to be where functional analysis really starts. But as it's one of those "there ...
- 26.8k
26
votes
7
answers
10k
views
Uniformly Sampling from Convex Polytopes
How to choose a point uniformly from a convex polytope $P \subset [0,1]^n$ defined by some inequalities, $Ax < b$? (Here $A$ is an $m \times n$ matrix, $x \in \mathbb{R}^n$, and $b \in \mathbb{R}^...
- 22.8k
8
votes
4
answers
530
views
Inside-out polygonal dissections
A dissection of a polygon $P$
is a partition of $P$ into a finite number of pieces, which can then be rearranged
(via planar translations and rotations) and joined (without overlap) to form a new ...
- 151k
8
votes
3
answers
1k
views
Ramanujan's Master Formula: A proof and relation to umbral calculus
The Ramanujan's master theorem states that:
$$
\int_0^{\infty}x^{s-1}\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}a_nx^ndx=\Gamma(s)a_{-s}
$$
I found a really strange proof recently on a personal blog:
Define
$...
- 302
4
votes
2
answers
341
views
Cutting convex regions into equal diameter and equal least width pieces - 2
This post is a spinoff from Cutting convex regions into equal diameter and equal least width pieces
Definitions: The diameter of a convex region is the greatest distance between any pair of points in ...
- 5,979
55
votes
6
answers
8k
views
Is it possible to partition $\mathbb R^3$ into unit circles?
Is it possible to partition $\mathbb R^3$ into unit circles?
- 1,414
44
votes
11
answers
26k
views
Algorithm for finding the volume of a convex polytope
It's easy to find the area of a convex polygon by division into triangles, but what is the optimal way of finding the volume of higher-dimensional convex bodies? I tried a few methods for dividing ...
- 441
40
votes
5
answers
10k
views
Is there a natural measures on the space of measurable functions?
Given a set Ω and a σ-algebra F of subsets, is there some natural way to assign something like a "uniform" measure on the space of all measurable functions on this space? (I suppose first ...
- 1,475
32
votes
6
answers
3k
views
Can distribution theory be developed Riemann-free?
I imagine most people who frequent MO have been indoctrinated into the point of view that the Riemann integral can be safely discarded once one has taken the time to develop the Lebesgue integral. ...
- 29.2k
27
votes
5
answers
3k
views
Nice applications for Schwartz distributions
I am to teach a second year grad course in analysis with focus on Schwartz distributions. Among the core topics I intend to cover are:
Some multilinear algebra including the Kernel Theorem and ...
26
votes
2
answers
4k
views
3D models of the unfoldings of the hypercube?
There are (apparently) 261 distinct unfoldings of the 4D hypercube, a.k.a., the
tesseract, into 3D.1
These unfoldings (or "nets") are analogous to the 11 unfoldings of
the 3D cube into the plane.2
...
- 151k
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
99
votes
7
answers
20k
views
Can we cover the unit square by these rectangles?
The following question was a research exercise (i.e. an open problem) in R. Graham, D.E. Knuth, and O. Patashnik, "Concrete Mathematics", 1988, chapter 1.
It is easy to show that
$$\sum_{1 \...
- 5,502
64
votes
6
answers
5k
views
Shortest closed curve to inspect a sphere
Let $S$ be a sphere in $\mathbb{R}^3$. Let $C$ be a closed curve in $\mathbb{R}^3$ disjoint from and
exterior to $S$
which has the property that every point $x$ on $S$ is visible to some point $y$ of $...
- 151k
36
votes
10
answers
6k
views
Determining a surface in $\mathbb{R}^3$ by its Gaussian curvature
A curve in the plane is determined, up to orientation-preserving
Euclidean
motions, by its curvature function, $\kappa(s)$.
Here is one of my favorite examples, from
Alfred Gray's book,
Modern ...
- 151k
28
votes
6
answers
12k
views
Almost orthogonal vectors
This is to do with high dimensional geometry, which I'm always useless with. Suppose we have some large integer $n$ and some small $\epsilon>0$. Working in the unit sphere of $\mathbb R^n$ or $\...
- 18.7k
26
votes
6
answers
8k
views
prime ideals in C([0,1])
It is clear that each maximal ideal in ring of continuous functions over $[0,1]\subset \mathbb R$ corresponds to a point and vice-versa.
So, for each ideal $I$ define $Z(I) =\{x\in [0,1]\,|\,f(x)=0, ...
- 5,063
23
votes
3
answers
6k
views
Density of smooth functions under "Hölder metric"
This question came up when I was doing some reading into convolution squares of singular measures. Recall a function $f$ on the torus $T = [-1/2,1/2]$ is said to be $\alpha$-Hölder (for $0 < \alpha ...
- 505
10
votes
0
answers
804
views
Topological dimension, Hausdorff dimension, and Lipschitz mappings
I can prove the following result. Here $\operatorname{dim} X$ stands for the topological dimension and $\mathcal{H}^n$ denotes the Hausdorff measure.
Theorem. Suppose that $f:\mathbb{R}^n\supset\...
- 28k
10
votes
2
answers
926
views
Is there a volume-preserving diffeomorphism of the disk with prescribed singular values?
This is a cross-post. While working on a variational problem, I have reached to the following question.
Let $0<\sigma_1<\sigma_2$ satisfy $\sigma_1\sigma_2=1$, and let $D \subseteq \mathbb{R}^2$...
- 6,741
7
votes
2
answers
1k
views
G-spaces and manifolds
In his book "The geometry of geodesics" H. Busemann defines the notion of a G-space to be a space which satisfies the following axioms:
The space is metric
The space is finitely compact, i.e., a ...
- 579
5
votes
1
answer
500
views
Hausdorff dimension of the graph of a BV function
Let $u: \Omega\subset \mathbb{R}^N \to \mathbb{R}^M$ be a $BV$ function.
Is the Hausdorff dimension of the graph of $u$ equal to $N$? How can we prove it?
Update.
In an answer to this post, it ...
- 839
102
votes
6
answers
11k
views
Is there an analogue of curvature in algebraic geometry?
I am not an expert, but there seems to be an enormous technical difference between algebraic geometry and differential/metric geometry stemming from the fact that there is apparently no such thing as ...
- 29.2k
100
votes
6
answers
5k
views
Light rays bouncing in twisted tubes
Imagine a smooth curve $c$ sweeping out a unit-radius disk that is
orthogonal to the curve at every point.
Call the result a tube.
I want to restrict the radius of curvature of $c$ to be at most 1.
I ...
- 151k
71
votes
16
answers
21k
views
Is there a nice application of category theory to functional/complex/harmonic analysis?
[Title changed, and wording of question tweaked, by YC, because the original title asked a question which seems different from the one people want to answer.]
I've read looked at the examples in most ...
60
votes
1
answer
7k
views
Probability that a stick randomly broken in five places can form a tetrahedron
Edit (June 2015): Addressing this problem is a brief project report from the Illinois Geometry Lab (University of Illinois at Urbana-Champaign), dated May 2015, that appears here along with a foot-...
- 7,831
58
votes
14
answers
19k
views
Open problems in Euclidean geometry?
What are some (research level) open problems in Euclidean geometry ?
(Edit: I ask just out of curiosity, to understand how -and if- nowadays this is not a "dead" field yet)
I should clarify a bit ...
51
votes
3
answers
3k
views
Can the sphere be partitioned into small congruent cells?
On the unit $2$-sphere ${\mathbb S}^2$ furnished with the geodesic distance, a subset homeomorphic to a planar disk is called a cell. A finite family of cells is a tiling if their interiors are ...
- 7,276
38
votes
1
answer
2k
views
Sofa in a snaky 3D corridor
What is the largest volume object that can pass though a
$1 \times 1 \times L$ "snaky" corridor, where $L$ is large
enough to be irrelvant, say $L > 6$.
...
- 151k
35
votes
6
answers
6k
views
How to explain the concentration-of-measure phenomenon intuitively?
One way to phrase the
"concentration-of-measure"
phenomenon is that,
for a Euclidean sphere $S^d$ in $d$ dimensions, for large $d$,
"most of the mass is close to the equator, for any equator."1
Q. ...
- 151k
34
votes
6
answers
8k
views
Covering a unit ball with balls half the radius
This is a direct (and obvious) generalization of the recent MO question, "Covering disks with smaller disks":
How many balls of radius $\frac{1}{2}$ are needed to cover completely a ball of ...
- 151k
30
votes
5
answers
16k
views
How to check if a box fits in a box?
How could I calculate if a rectangular cuboid fits in an other rectangular cuboid, it may rotate or be placed in any way inside the bigger one.
For example would, (650,220,55) fit in (590,290,160), ...
- 429
28
votes
8
answers
5k
views
Convex hull in CAT(0)
Let $X$ be complete $\mathop{CAT}(0)$-space and $K\subset X$ be a compact subset.
Is it true that convex hull of $K$ is compact?
Comments:
Convex hull of $K$ = intersection of all closed convex sets ...
- 45k
25
votes
2
answers
2k
views
Functional approach vs jet approach to Lagrangian field theory
Context: I am a PhD student in theoretical physics with higher-than-average education on differential geometry. I am trying to understand Lagrangian and Hamiltonian field theories and related concepts ...
- 3,292
25
votes
1
answer
7k
views
Hanging a ball with string
What is the shortest length of string that suffices to hang
a unit-radius ball $B$?
This question is related to an earlier MO question, but I think different.
Assume that the ball is frictionless.
...
- 151k
24
votes
3
answers
3k
views
Integer-distance sets
Let $S$ be a set of points in $\mathbb{R}^d$; I am especially interested in $d=2$.
Say that $S$ is an integer-distance set if every pair of points in $S$ is separated
by an integer Euclidean distance.
...
- 151k
23
votes
2
answers
3k
views
States in C*-algebras and their origin in physics?
in $C^*-$algebras with unit element, there is the definition of a state, as a functional $\omega$ with $\omega(e)=||\omega||=1.$
Now, of course there is also in classical physics and quantum ...
- 243
17
votes
2
answers
5k
views
Square of the distance function on a Riemannian manifold
Let $(M^n,g)$ be a smooth Riemannian manifold. Consider the square of the distance function
$$dist^2\colon M\times M\to \mathbb{R}$$
given by $(x,y)\mapsto dist^2(x,y)$. It is easy to see that this ...
- 21.8k
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