All Questions
4,828 questions
17
votes
1
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When is the Gromov--Hausdorff limit of a sequence of manifolds itself a manifold?
Suppose a metric space $X$ is the Gromov--Hausdorff limit of a sequence of Riemannian three-manifolds $M_i$ with Ricci curvature bounded from below and not collapsed. Is $X$ a manifold? What ...
- 201
17
votes
3
answers
3k
views
Nonseparable example in dimension theory?
Could you give me an example of a complete metric space with covering dimension $> n$ all of which closed separable subsets have covering dimension $\le n$?
The question closely related to this ...
- 1,785
17
votes
3
answers
922
views
Random permutations from Brownian motion
Let $B(t)$ be a Brownian motion. The ordering of $(0, B(1), ..., B(n-1)) $ is a random permutation in $S_n$. This is not uniform for $n>2$ since the probabilities of the identity permutation $[123.....
- 28k
17
votes
1
answer
1k
views
Tropical mathematics and enriched category theory
Is there a connection between tropical mathematics and the Lawvere enriched category theory approach to metric spaces? I guess I will give a partial answer to this below, but I mean can they be ...
- 1,271
17
votes
1
answer
390
views
17
votes
0
answers
1k
views
Almost monochromatic point sets
There are many sort of equivalent theorems about monochromatic configurations in finite colorings, such as Van der Waerden, Hales-Jewett or Gallai's theorem, the latter of which states that in a ...
- 18.7k
17
votes
0
answers
224
views
GPS calculations under $L^p$ norms
GPS calculations require finding a sphere externally tangent to
four given spheres, an
Apollonian problem
in $\mathbb{R}^3$.
The center of that fifth sphere is one of the $16$ possible solutions to
...
- 151k
17
votes
0
answers
488
views
Large almost equilateral sets in finite-dimensional Banach spaces
Question: Does there exist a function $C:~(0,1)\to
(0,\infty)$ such that for each $\varepsilon\in(0,1)$ every Banach space
$X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point
set $\{x_i\...
- 4,878
17
votes
0
answers
731
views
Does every connected set that is not a line segment cross some dyadic square?
A dyadic square is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists a connected subset ...
- 171
16
votes
3
answers
1k
views
Can a convex polytope with $f$ facets have more than $f$ facets when projected into $\mathbb{R}^2$?
Let $P$ be a convex polytope in $\mathbb{R}^d$ with $n$ vertices and $f$ facets.
Let $\text{Proj}(P)$ denote the projection of $P$ into $\mathbb{R}^2$.
Can $\text{Proj}(P)$ have more than $f$ facets?
...
- 163
16
votes
3
answers
2k
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 ...
- 151k
16
votes
6
answers
3k
views
Smallest area shape that covers all unit length curve
On a euclidean plane, what is the minimal area shape S, such that for every unit length curve, a translation and a rotation of S can cover the curve.
What are the bounds of the shape's area if this ...
- 613
16
votes
3
answers
1k
views
Ultraproducts of Banach spaces versus model theoretic ultraproduct
Reading about ultraproducts in model theory and in Banach spaces leads to two distinct definitions. E.g., for an ultrapower given by an ultrafilter $\mu$ on $\mathbb{N}$, both notions of ultrapower ...
16
votes
2
answers
1k
views
Algebraic surface of constant width?
Does there exist an irreducible polynomial $f \in \mathbb{R}[x, y, z]$ such that:
$$ V := \{ (x, y, z) \in \mathbb{R}^3 : f(x, y, z) \leq 0 \} $$
is a solid of constant width with a finite symmetry ...
- 12.4k
16
votes
4
answers
3k
views
covering by spherical caps
Consider the unit sphere $\mathbb{S}^d.$ Pick now some $\alpha$ (I am thinking of $\alpha \ll 1,$ but I don't know how germane this is). The question is: how many spherical caps of angular radius $\...
- 96.4k
16
votes
5
answers
1k
views
A characterization of convexity
While doing some research on polytopes I came to the following question. Maybe it's already somewhere but anyway I'll post it here.
Let $X\subset \mathbb{R}^3$ be such that, for every plane $P$, $P\...
- 1,001
16
votes
5
answers
903
views
Which metric spaces have this superposition property?
Let $A \subset X$ and $B \subset X$ be two isometric subsets of a metric space $X$. So there is an isometry $f: A \to B$.
Say that a metric space $X$ has the superposition property (my terminology) ...
- 151k
16
votes
4
answers
2k
views
Neusis constructions
Is there some simple description of which complex numbers are "constructible" with straightedge and compass and neusis?
See http://en.wikipedia.org/wiki/Constructible_number and http://en.wikipedia....
user5810
16
votes
5
answers
717
views
Minimal blocking objects with shadows like a cube
This is a more geometric version of the previous question,
"Lattice-cube minimal blocking sets". I will first specialize to $\mathbb{R}^3$, $d=3$.
View an $n \times n \times n$ cube $C_3(n)$ as ...
- 151k
16
votes
4
answers
2k
views
Point sets in Euclidean space with a small number of distinct distances
It is well known and not hard to prove that the regular simplex in n-dimensions is the only way to place n+1 points so that the distance between distinct pairs of points is always the same. My general ...
- 1,314
16
votes
1
answer
1k
views
Is the "equidistant curve" to an algebraic curve algebraic?
Let $ L \subseteq \mathbb{R}^2 $ be a smooth real algebraic curve. Let's fix some parameter $ \delta \in \mathbb{R} $ and for every point $ (x,y) \in L $ define $$ L_{\delta}(x,y) = (x,y) + \delta n(x,...
- 317
16
votes
2
answers
626
views
Cantor-Bernstein for quasi-isometric embeddings?
Suppose that two finitely generated groups quasi-isometrically embed into each other. Does it follow that the two groups are quasi-isometric? Recall that a quasi-isometry is a quasi-isometric ...
- 11.1k
16
votes
2
answers
1k
views
Integer lattice points on a hypersphere
Is the following statement true?
For every integer $n\ge2$ and every integer $k\ge0$ there exists a hypersphere in $\mathbb{R}^n$ (circle, sphere etc) containing exactly $k$ integer lattice points ...
- 513
16
votes
2
answers
2k
views
There are two points on the Earth's surface that ... ?
At every moment in time, there are two points on the Earth's surface that have the same $\lbrace x, y, z, ... \rbrace$...?
What is the strongest, most impressive statement one can make here? The ...
- 151k
16
votes
3
answers
676
views
Are there two deformations of a wire loop such that neither can pass through the other?
My 10-year old son asked me this simple question, and I've been unable to answer it.
Suppose we start with two (unlinked) circular wire loops (maybe different sizes), and allow both to be ...
- 343
16
votes
1
answer
888
views
Kakeya crossed-needles problem
The Kakeya needle problem asks for the
minimum area planar region in which one can completely turn around a line segment through
a series of translations and rotations. There is no minimum: There are &...
- 151k
16
votes
1
answer
1k
views
Mapping class group and CAT(0) spaces
I hope the questions are not too vague.
Is the mapping class group of an orientable punctured surface $CAT(0)$ ?
Is any of the remarkable simplicial complexes (curve complex, arc complex...) built ...
- 828
16
votes
2
answers
5k
views
Weighted area of a Voronoi cell
Let $X = \{ x_1,\dots,x_n\} $ denote a set of $n$ points in the unit square $S = [0,1]\times[0,1]$, and let $w = \{w_1,\dots,w_n\}$ denote a set of weights corresponding to the $n$ points in $X$. ...
- 195
16
votes
3
answers
2k
views
A random walk on random lines
I am wondering if this random walk remains finite with positive probability.
Start with three lines $A,B,C$ that are extensions of an equilateral triangle.
Let $p_0$ be one corner. Generate a line $...
- 151k
16
votes
2
answers
466
views
Does a certain points and lines configuration exist?
For which $n$ we may mark $n$ red and $n$ blue points on the Euclidean plane, not all on a line, so that any line which passes through two points of different colour contains another point?
For $n=...
- 109k
16
votes
1
answer
589
views
(A question about)${}^3$ 3-dimensional convex bodies
Related to the questions mathoverflow.net question No. 137850 and mathoverflow.net question No. 39127, is there a 3-dimensional convex body other than a ball whose perpendicular projections in all ...
- 7,276
16
votes
2
answers
756
views
Can a sphere glued into a soft 3d-mattress rotate continuously? (manifolds, SU(2) and the belt trick)
The question is triggered by the wonderful animations by Jason Hise:
https://www.youtube.com/watch?v=LLw3BaliDUQ
https://www.youtube.com/watch?v=6Ul_-ABYaYU
https://www.youtube.com/watch?v=...
16
votes
1
answer
1k
views
Does Gromov's Waist Inequality imply Borsuk-Ulam?
I'm curious if anyone can see a route to get the Borsuk-Ulam theorem from Gromov's waist inequality. For the sake of notation, here's the inequality:
Let $S^n$ denote the round unit sphere in $\...
- 647
16
votes
3
answers
2k
views
Expected Degree of a vertex in Delaunay Triangulations
Assume you have a Poisson point process of constant intensity $\lambda$ in the Euclidean plane. From this point process we construct the Delaunay triangulation (or the Voronoi tessellation for that ...
- 3,626
16
votes
2
answers
1k
views
Are Penrose tilings universal? Do aperiodic universal tilings exist?
Consider a tiling of the plane using tiles of at least two types (e.g, a Penrose tiling such as that shown at the bottom of this question, which tiles the plane with two types of tiles). List the tile ...
- 4,922
16
votes
2
answers
1k
views
Maximum area of the intersection of a parallelogram and a triangle
How large can the intersection of a parallelogram (or a square, if you prefer) with a triangle be, if each of them is of unit area? It is easy to see that the intersection can be of area 3/4 – is this ...
- 7,276
16
votes
1
answer
985
views
Generalizing the Mazur-Ulam theorem to convex sets with empty interior in Banach spaces
The Mazur-Ulam theorem (1932) states that any isometry of a normed linear space is affine. See Nica (Expo. Math. 30 (2012), 397-398; arXiv:1306.2380) for a very elegant proof.
Question: Let $M$ be a ...
- 4,878
16
votes
1
answer
1k
views
Random polycube shapes
I am wondering if it is hopeless to obtain any firm results
on the following model of a "random polycube shape."
First, a polycube in $\mathbb{R}^3$
is a connected face-to-face gluing of unit cubes.
(...
- 151k
16
votes
1
answer
533
views
Is there a degenerate simplex in $\mathbb{R}^{8 k-1}$ with odd integer edge lengths?
The Cayley-Menger determinant gives the squared volume of a simplex in $\mathbb{R}^n$ as a function of its $n(n+1)/2$ edge lengths:
$$v_n^2 = \frac{(-1)^{n+1}}{(n!)^2 2^n}
\begin{vmatrix}
0&d_{01}^...
- 2,902
16
votes
1
answer
774
views
Minimizing the excursion of a sum of unit vectors
I have $n$ unit-length vectors $v_i$ in $\mathbb{R}^3$, whose
sum is zero:
$$ v_1 + v_2 + \cdots + v_n = 0 \; .$$
Now I form the closed polygon $P$ in space by placing them head to tail.
So the ...
- 151k
16
votes
2
answers
590
views
Can you perturb an inscribed polytope so all its edges grow?
Consider the family of convex simplicial polytopes with vertices in the unit sphere of $\mathbb{R}^n$ which have the origin as an interior point.
My question is the following:
Let $P, P'$ be two non-...
16
votes
2
answers
731
views
A reference to a characterization of metric spaces admitting an isometric embedding into a Hilbert space
I am looking for a reference to the bipartite version of the Schoenberg's criterion of embeddability into a Hilbert space. The Schoenberg criterion is formulated as Proposition 8.5(ii) of the book &...
- 41.8k
16
votes
1
answer
537
views
Balls in Hilbert space
I recently noticed an interesting fact which leads to a perhaps difficult question. If $n$ is a natural number, let $k_n$ be the smallest number $k$ such that an open ball of radius $k$ in a real ...
- 2,049
16
votes
1
answer
2k
views
Origami Constructions: Intersecting two Circles
It is well known that every construction that can be performed with compass and straightedge alone can also be performed using origami, see:
R. Geretschlager. Euclidean Constructions and the Geometry ...
- 529
16
votes
1
answer
444
views
Is there a conceptual reason why so many triplets of lines in a triangle are concurrent?
One of the striking phenomena one can't help but notice in elementary Euclidean geometry is how easy it appears to be to define triples of lines in a triangle which meet in a point. Now for each ...
- 32.5k
16
votes
1
answer
511
views
Subdividing a polyhedral space into convex simplices
A (Euclidean) polyhedral space is a metric space obtained by "gluing together" several (let's assume finitely many) Euclidean simplices (of varying dimensions) by identifying some faces via isometries....
- 32.4k
16
votes
1
answer
508
views
Can solutions to Thomson's problem have pentagons?
Thomson's problem asks for the minimum-energy configuration for $N$ electrons on a sphere (refs: https://en.wikipedia.org/wiki/Thomson_problem, https://sites.google.com/site/adilmmughal/...
- 1,203
16
votes
3
answers
1k
views
Smallest square to wrap a cylinder
Suppose you need to gift-wrap a cylinder (e.g., a can of tennis balls, or a large candle)
of height $h$ and radius $r$.
Here wrap is the natural sense of covering the surface area of the cylinder ...
- 151k
16
votes
1
answer
806
views
Blocking visibility with cylinders
Suppose you have a supply of infinite-length, opaque, unit-radius cylinders,
and you would like to block all visibility from a point
$p \in \mathbb{R}^3$ to infinity with as few cylinders as possible.
...
16
votes
1
answer
883
views
Hearing the 17 planar symmetry groups
Though I'm sure it's not really hard to work out for myself, does anyone know a reference for the spectra of the Laplacian on the 17 flat compact orbifolds that underlie the 17 planar symmetry groups. ...
- 17.6k