Euclidean, hyperbolic, discrete, convex, coarse geometry, comparisons in Riemannian geometry, symmetric spaces.
3
votes
0answers
24 views
Which unfoldings of the hypercube tile 3-space: How to check for isometric space-fillers?
Recently Mark McClure constructed and displayed
the 261 unfoldings of the hypercube (tesseract)
in response to the question,
"3D models of the unfoldings of the hypercube?":
The first 9 unfoldings ...
3
votes
2answers
89 views
regular polyhedra (and polytopes) in hyperbolic geometry, and generalisations
While there exist regular tesselations of the hyperbolic plane with arbitrary regular polygons, there are no new regular polyhedra in hyperbolic (3D) space. This being quite trivial, it is probably ...
4
votes
0answers
50 views
Can the GUE be thought of as a uniform point in a high-dimensional polytope
I have thought about this question for a long time and could only find partial answers.
The Gaussian Unitary Ensemble (or GUE) is the eigenvalues of a random Hermitian matrix with complex Gaussian ...
4
votes
0answers
133 views
When is a word metric on a CAT(-1) group a bounded distance from the orbit map of an isometric action on some CAT(-k) metric space?
Let $\Gamma$ be a group admitting a discrete and cocompact action on a CAT(-1) space.
Let $d$ a word metric on $\Gamma$ coming from some finite set of generators.
My question is:
Does there exist a ...
8
votes
1answer
197 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
...
-1
votes
0answers
40 views
Orthogonal vector to arbitrary vector in R3 [closed]
I got a vector $(0, 0, 0)^T \neq v \in R^3$. Now I want a closed formula for some orthogonal vector to $w$ (I don't care which).
My problem is that if I, for instance, fix $w_1$ and $w_2$ then $w_3 = ...
1
vote
0answers
39 views
How often can a single length occur as a boundary distance?
Given a bounded domain $\Omega\subset\mathbb R^n$ ($n\geq2$), how often can a single real number $r>0$ appear as a distance of two points on $\partial\Omega$?
We can make any assumptions about the ...
3
votes
0answers
100 views
How to build the smallest regular n-sided polygon that covers an (n-1)-sided polygon? [migrated]
I want to build a figure that contains seven regular polygons, from a triangle up to a nonagon, where each n-sided polygon covers, with the minimal area possible, the n-1 sided one. An added ...
7
votes
0answers
102 views
Ricocheting pinball-like shot: Complexity?
Suppose one has $n$ perfect two-sided mirror segments in the plane $\mathbb{R}^2$.
The segments are open, excluding their endpoints.
They are disjoint as closed segments, i.e., no pair shares an ...
5
votes
1answer
262 views
Geodesics on manifolds with boundary
Let $(M,g)$ be a Riemannian manifold with non-empty boundary. Is there any notion of injectivity radius on $(M,g)$ in points away from the boundary? By this I mean points lying in $M- \partial M$. ...
2
votes
0answers
99 views
How far can I push a ball of radius r into the corner of a convex polytope in d-dimensions?
Let $K \subset \mathbb{R}^d$ be a convex polytope and let $B(x,r) = \{y \in \mathbb{R}^d \,:\, \Vert y-x \Vert_2 \leq r\}$ be the ball of radius $r$ centered at $x$.
For any radius $r \geq 0$ let ...
-1
votes
0answers
49 views
Quotient of Euclidean space under the action of a permutation group
Suppose that $G$ is a permutation group acting on the Euclidean space $\mathbb{R}^n$. I am interested in the geometric properties of the quotient space $\mathbb{R}/G$. Since $G$ is finite and the ...
2
votes
1answer
194 views
An identity for Futaki-Donaldson invariant
Let $(X,L)$ be a polarized projective variety
Given an ample line bundle $L\to X$, then a test configuration for the pair $(X,L)$ consists of :
a scheme $\mathfrak X$ with a $\mathbb C^*$-action
a ...
2
votes
2answers
212 views
Length of non-horizontal curve
Let $M$ be a sub-Riemannian space.
Consider a smooth curve $\gamma:[0,1]\to M$ such that
$\dot\gamma(t)\not\in H_{\gamma(t)}$, where $H_{\gamma(t)}$ is the horizontal subbundle ( i.e. $\gamma$ is ...
2
votes
1answer
78 views
Is there a name for the level-sets of the signed distance function to a set in a metric space?
$\newcommand \X {\mathcal{X}}$
$\newcommand \sd {d_{\rm sign}}$
Let $(\X, d)$ be a metric space and define the distance between a point $x \in \X$ and a set $S \subset X$ by $d(x,S) = \inf_{y \in S} ...
9
votes
1answer
138 views
Optimal shape for stabbing balls in $\mathbb{R}^3$
I have radius $r < \frac{1}{2}$ congruent balls with centers randomly distributed uniformly within a region,
say, within a unit-radius sphere $S$.
I shoot a ray/path through $S$, hoping to ...
3
votes
0answers
232 views
Help in how to estimate a parameter using Poisson point process
Let $\mathbf{e_0,e_1,\ldots,e_n}$ $\in$ $R^d$ denote points of a random Poisson point process in $R^d$. which is centered so that $e_0 =0$.
Considering nearest neighbor distances: for a specific ...
3
votes
1answer
103 views
Distance function from a topological submanifold
Let $(M,g)$ be a Riemannian manifold, and let $N\subset M$ be an embedded sphere that is everywhere smooth except for a single point at which the embedding will only be $C^0$.
How much regularity can ...
2
votes
1answer
139 views
Lorentzian metrics on the torus up to continuos deformations
Any two Riemannian metrics can easily be deformed into each other, only obtaining positive definite metrics in between.
However, for metrics of other signatures this might not be possible.
Which ...
1
vote
0answers
92 views
Virtually abelian centralizers
This is a sort of a follow-up question to Limits of conjugated subgroups (though it might not seem at first glance to have much to do with it.)
Anyway, I'm wondering what sort of groups have the ...
1
vote
0answers
41 views
Why finite dimensional MCS-space implies $\mathbb{R}^m\times cone$ locally?
Frequently, I see a statment of the result in reference that a finite dimensional Alexandrov spaces is locally a product $\mathbb{R}^m\times cone$. It seems that this result comes from Perelman's ...
0
votes
1answer
98 views
Convergence in the Wasserstein metric and the square root function
Let $f$ be a smooth probability distribution on the unit square $S$ such that $f(x)>0$ on $S$. Let $\{g_i\}$ be a sequence of smooth probability distributions such that $g_i(x)>0$ on $S$ as ...
8
votes
0answers
118 views
Closed geodesics avoiding points in hyperbolic surfaces
Let $\Sigma$ be a closed hyperbolic surface. Is it true that for any finite collection of points $x_1,\ldots,x_n\in\Sigma$ there exists a closed geodesic $\gamma$ containing none of them?
Remark: It ...
4
votes
0answers
184 views
How many points does 'the-most-point-contained-circle' contain at least?
Question : Given any $n$ distinct points $S$, consider the $\binom n2$ discs $D_{pq}$ formed by picking a pair of points $p,q$ from $S$ and using them as a diameter. For each disc $D_{pq}$, let ...
15
votes
2answers
388 views
Can all unit-distance graphs have their vertices at algebraic integers?
A graph $G$ is described as a unit-distance graph if there exists a function $f:G \rightarrow \mathbb{C}$ such that for every edge $(u,v) \in E(G)$, we have $|f(u) - f(v)| = 1$.
Obviously, we can ...
7
votes
0answers
93 views
Approximate singular value decomposition in Banach spaces
I am interested in generalisations to Banach spaces of the following construction, which relates to the singular value decomposition of a finite-dimensional linear map. If $V$, $W$ are ...
7
votes
1answer
227 views
The Minkowski sum of two curves
Let $\gamma$ be a continuous curve in the complex plane without self-intersections and let $\lambda$ be a complex non-real number less than 1 in modulus. Put $\gamma'=\lambda\gamma$.
Question. Is it ...
1
vote
1answer
289 views
Does the singular cohomology for a metric space of finite topological dimension vanish in high dimensions?
It is known that by applying the universal coefficient theorem, the singular cohomology of closed manifold with coefficient $\mathbb{Z}_2$ vanishes in high dimensions. But for a metric space $M$ with ...
10
votes
1answer
270 views
Soft question: mathematics about truchet tiles
It seems that this is the first question on Truchet tiles on MO.
Shown above is a picture of a random tile, which you can see the resulting configuration is much like many membranes of cells.
I ...
1
vote
1answer
88 views
Convex subcomplexes of CAT(0) cubical complexes
Is the following statement true? If so, can anyone provide a reference?
Let $X$ be a CAT(0) cubical complex, and let $Y$ be a connected
subcomplex of $X$. Then the following are equivalent:
...
2
votes
1answer
89 views
Parallel transport on a Hadamard manifold
Suppose, $X$ is a Hadamard manifold, i.e., a simply connected manifold of non-positive sectional curvature. Fix a point $w$ in $X$. Consider any three points $x, y, z$ in $X$. Let $\tau_{x, w}$ and ...
3
votes
1answer
72 views
conjugacy between geodesic flows on 2-tori
Let $(T_1,g_1)$ and $(T_2,g_2)$ be two flat tori of dimension 2 such that their geodesic flows are $C^0$-conjugated, is there an isometry between $(T_1,g_1)$ and $(T_2,g_2)$ ?
I emphasize the fact ...
6
votes
1answer
146 views
Embedding Euclidean buildings into products of trees
A Euclidean building has a natural metric space structure. (A definition of Euclidean building can be found on Wikipedia, or, more expansively, in Section 4 of Kleiner-Leeb.)
Question: Is it true ...
0
votes
0answers
32 views
Covering number of the range of a function
I have come across the need to know a bound on a certain curious quantity: the covering number of the range of a continuous function $f: D \rightarrow \mathbb{R}^n$, where $D \subseteq \mathbb{R}^m$. ...
6
votes
1answer
256 views
Which surfaces have only a finite number of connecting geodesics?
Q1. For a smooth, closed (compact) surface $S$ embedded in $\mathbb{R}^3$,
under which conditions is it true that, for every pair of points
$a,b \in S$, there are an infinite number of ...
10
votes
1answer
433 views
Metric $d(A,B) = \mathbb P(\overline A\cup\overline B\mid A\cup B)$
I'm wondering where the relative probabilistic distance was first studied:
$$d(A,B) =\mathbb P(\overline A\cup\overline B\mid A\cup B)$$
where $\overline A$ is the complement of $A$.
A web search ...
0
votes
1answer
78 views
Practical Algorithm for Comparing the Discrepancy of Point Sets (on Unit Hyper Spheres)
I have devised a simple geometric algorithm for generating a sequence of points on unit hyper spheres; that algorithm depends on a single real parameter, which I would like to optimize in order to get ...
8
votes
1answer
109 views
Tilting the $d$-cube to vertically separate its vertices
Let $C_d$ be a unit edge-length cube in $d$ dimensions.
I would like to orient it ("tilt" it) so that the vertical (last) coordinates
of its $2^d$ vertices are maximally separated, in the sense
that ...
2
votes
1answer
206 views
Given a set of 2D vertices, how to create a minimum-area polygon which contains all the given vertices?
Not sure whether this question belongs here or math.stackexchange.
You can assume that all the vertices are unique. The given vertices can be the vertices of the polygon, thus they do NOT have to be ...
6
votes
3answers
419 views
Polynomial threading through a monotone corridor
I have a need to find a polynomial of minimal degree that connects
two points and stays within a given
"corridor," by which I mean an $x$-monotone polygon.
Here is an example:
...
4
votes
1answer
485 views
Focus of parabola using only a ruler
It is an easy exercise that using ruler and compass one find the focus of a given parabola.
Can one do the same using only a ruler? -- if not, why?
1
vote
0answers
103 views
Shortest rope to capture a sphere of diameter 1 [duplicate]
I have a perfect rigid sphere of diameter 1.
I have infinite supply of rope. The rope is infinitely flexible and can be cut or glued without losing or adding length. The rope can be glued at any ...
3
votes
1answer
161 views
The relation between Hausdorff dimension of an $n$-manifold and $n$
It is known that for a topological space with different metrics, the Hausdorff dimensions may not be equal in general.
For the case of manifolds, suppose $M$ is a $n$-manifold with a ...
1
vote
0answers
44 views
Length invariance under nondecreasing changes of parameters
Suppose that $f\colon [0,1]\to [0,1]$ is a continuous, surjective and nondecreasing function, for example the Cantor function. Let $X$ be a metric space (not necessarily a length space) and let $L$ be ...
3
votes
1answer
281 views
The Praying Eyes theorem generalized
Is there an obvious way to slice two spheres (no necessarily equal)
simultaneously such that the sections share common areas? I mean, I cannot see another way (easier) different from the way provided ...
17
votes
1answer
233 views
1
vote
2answers
194 views
Regular paths along surface of sphere
I'm trying to create a program where a small ball is supposed to move along the surface of a sphere, which is given by its radius $r$ and the center $c$.
The movement should be repetitive, so that ...
11
votes
1answer
282 views
Are all well behaved “mean” functions on $\mathbb{R}^+$ equivalent?
Given a set $S$, a function $M: S\times S \rightarrow S$ is a mean if it satisfies the properties:
$M(a,a)=a\qquad$ (identity)
$M(a,b)=M(b,a)\qquad$ (commutativity).
and possibly
...
4
votes
2answers
322 views
Distance function to a submanifold
Let $M$ be a compact Riemannian manifold and $\Sigma\subset M$ a closed submanifold. Given $x\in M$ we define the distance function to $\Sigma$ by $$d_\Sigma(x):=\inf\{d(x,y):y\in \Sigma\},$$ where ...
6
votes
0answers
78 views
Unbalanced equipartitions
Let $K$ be a compact convex set in the plane.
Say that a perimeter-halving partition of $K$
is a partition of $K$
into two pieces by a chord (a segment with endpoints
on the boundary $\partial K$) ...
