Euclidean, hyperbolic, discrete, convex, coarse geometry, comparisons in Riemannian geometry, symmetric spaces.

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3
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2answers
49 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
45 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 ...
2
votes
0answers
123 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
150 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
37 views

Orthogonal vector to arbitrary vector in R3 [on hold]

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 ...
0
votes
0answers
105 views

Construction of Penrose Tiling (P3) from Wieringa Roof [closed]

I am trying to find sources detailing the construction of the Wieringa* Roof with the purpose of projecting the roof onto the plane and generating a Penrose Tiling (P3). The method can be seen in ...
7
votes
0answers
99 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
261 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
136 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
229 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
40 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
183 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
268 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
87 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
255 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
108 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
481 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
160 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
281 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
321 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 $\partial K$) ...