Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,263
questions
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How do we calculate the gradient of this function defined using the Riemannian logarithm on a Riemannian manifold?
We consider the following function $\psi$ on an open subset $V\subset M,$ a Riemannian manifold of dimension $m,$ so that $\exp_p:U\to V$ is a diffeomorphism with its inverse $\log_p: V\to U$. Let $v\...
- 1,417
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165
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Convex hull of 3 points in Cartan-Hadamard manifolds
Can the convex hull of $3$ points in a Cartan-Hadamard manifold be smooth?
A Cartan-Hadamard manifold $M$ is a complete simply connected manifold with nonpositive curvature (so it includes the ...
- 7,047
4
votes
2
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240
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Convergence of metric spaces of increasing dimension
Given two metric spaces we can define the Gromov-Hausdorff (GH) distance. There are compactness results stating that a sequence of manifolds of a fixed dimension, with a uniform lower Ricci bound and ...
- 41
2
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103
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To find the convex planar region minimizing diameter when area and perimeter are given
The basic question is to find that planar convex region for which diameter is a minimum when area and perimeter are specified.
A partial answer is given here: http://nandacumar.blogspot.com/2012/11/...
- 5,493
14
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376
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Will a unit disk be completely covered by randomly placed disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ with probability $1$?
On a "bottom" disk of area $\pi$, we place "top" disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ such that the centre of each top disk is an independent uniformly random ...
- 2,653
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98
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Is every finite metric space representable in a pseudo-Euclidean space?
Let $X$ be a finite set with a (true) metric $d$ and $|X| = n$. Does there exist a set $Y$ of $n$ points in $R^n$ with a pseudo-Riemannian metric with signature $(n - k, k, 0)$ for some integer $k$ ...
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Question about $n$ random points in a regular polygon, and a limiting probability
Suppose we choose $n$ uniformly random points in a disk, then draw the smallest circle that encloses all of those points. There is evidence suggesting that the probability that the enclosing circle is ...
- 2,653
7
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2
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444
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Amalgamated product acting on CAT(0) cube complex
I was reading the following result from the book Metric spaces of non-positive curvature by Bridson and Haefliger.
Result:
Let $F_0,F_1$ and $H$ be groups acting properly
by isometries on complete $...
- 339
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105
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Intuition behind right-inverse of map from Johnson-Lindenstrauss Lemma
The Johnson–Lindenstrauss lemma states that for every $n$-point subset $X$ of $\mathbb{R}^d$ and each $0<\varepsilon\le 1$, there is a linear map $f:\mathbb{R}^d\to\mathbb{R}^{O(\log(n)/\varepsilon^...
- 4,969
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A gerrymandering problem - can you always turn a tie into a landslide victory?
Note: Here we use $|A|$ to denote the Lebesgue measure of a measurable subset $A$ of $\mathbb R^2$.
Your party is running for election! In your country, voters are approximately uniformly distributed. ...
- 4,832
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391
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Smallest sphere containing three tetrahedra?
What is the smallest possible radius of a sphere which contains 3 identical plastic tetrahedra with side length 1?
- 111
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113
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Binary codes with upper and lower bound on pairwise distance
The Gilbert-Varshamov bound provides a lower bound for codes of length $n$ with minimum pairwise distance (say $\frac{n}8$). If we wish for the codes to also have pairwise distances bounded above (say ...
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149
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Upper bound of special Cheeger constant on $(S^2,g)$
$(S^2,g)$ is 2-dimensional sphere with Riemannian metric.The Cheeger constant of $(S^2,g)$ is
$$
h(S^2,g)=\inf_{\gamma} \frac{|\gamma|_g}{\min\{|A_1|_g, |A_2|_g\}}
$$
take the infimum over all closed ...
- 165
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How does the extra rope length of this link/tangle scale with the inner triangle size?
The symmetric chiral link made of three long intertwined/linked/tangled flexible ropes of radius 1 shown in the figure, whose 6 ends all lie in a plane at spatial infinity and which are pulled ...
- 9
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Planar sections of convex sets in Cartan-Hadamard manifolds
Let $X$ be a convex set in Euclidean space $\mathbf{R}^n$ and $p\in\mathbf{R}^n$ be a fixed point. Then any plane $\Pi$ passing through $p$ intersects $X$ in a convex set. Conversely, this property ...
- 7,047
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An order statistics problem with some interesting geometry
Let $a_n$ be a given sequence of positive numbers, and $X_n$ a sequence of independent random variables with each $X_n$ uniformly distributed on $[0, a_n]$.
Question: Let $N \geq 2$ be an arbitrary ...
- 4,832
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1
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268
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If $\mathcal{H}^{n-1}(E)=0$ then $\mathbb{R}^n\setminus E$ is connected
Let $E\subset \mathbb{R}^n$ be a (measurable) subset with $\mathcal{H}^{n-1}(E)=0$, where $\mathcal H^{n - 1}$ is the ($n - 1$)-dimensional Hausdorff measure. I want to know if $\mathbb{R}^n\setminus ...
- 1,037
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131
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Need help understanding the geometry of a particular building structure
$\DeclareMathOperator\SL{SL}$I’m not primarily a geometer, so apologies if this question is worded poorly. I’ve been looking at asymptotic cones of connected semisimple Lie groups with at least one ...
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47
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Optimal packing and covering of a triangle with squares
We continue from Another variant of the Malfatti problem.
Given a triangle T and a number n, how to cover it with n squares (of possibly different dimensions) such that the sum of the areas (...
- 5,493
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145
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Arcular triangle inequality
Is it true that if inside a circular segment $S$, with vertices $a$ and $b$, we take two circular arcs, one from $a$ to $c$ and the other from $c$ to $b$, then the sum of the lengths of these two arcs ...
- 18.4k
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222
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A property for maps between metric spaces
Let $X, Y$ be metric spaces with distance functions denoted by $d_X, d_Y$ respectively. Consider a map $f \colon X \rightarrow Y$. I am interested in the following property: for every $x,y,z \in X$, ...
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What is the state of progress on this problem about continuous functions from spheres to Euclidean space?
In the 1954 paper Continuous Functions From Spheres to Euclidean Spaces, author Chung-Tao Yang cites the following problem:
Problem 1: Given a (continuous) map $f$ of an $(m+n-2)$-sphere $S^{m+n-2}$ ...
- 2,298
16
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560
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Snakes on a plane
A sleeping bag for a baby snake in $d$ dimensions (no, really) is a subset of $\mathbb{R}^d$ which can cover (via translation and rotation) every (piecewise-smooth for concreteness) curve of unit ...
- 19.2k
1
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208
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Qualitative values between two electrons in an atom or how to interpret these values?
This question is a little bit trying to understand physics through geometry of simplex:
Let $E_{i,j}$ be the ionization energy in times the number of hydrogen ionization energy for an element with ...
- 2,462
2
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0
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77
views
Another variant of the Malfatti problem
We try to add to A Variant of the Malfatti Problem
As stated in the Wikipedia entry on Malfatti circles, it is an open problem to decide, given a number $n$ and any triangle, whether a greedy method ...
- 5,493
4
votes
2
answers
190
views
Algorithm for grouping tetrahedra from Voronoi diagram
I have a set of 3D Voronoi generator points and their neighbouring points, which, when connected, should result in a Delaunay tetrahedralization. However, I'm having a hard time implementing this. My ...
- 142
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123
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Sufficient condition for existence of a closest-point projection from a neighborhood onto a subset of a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold and let $N$ be a subset of $M$.
On one hand, it is well known that if $N$ is an embedded submanifold of $M$, then it admits a tubular neighborhood, and, ...
- 134
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92
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Relation of geometric and polyhedral convergence
By Proposition 3.10(i) of Jorgensen and Marden's 1990 Algebraic and geometric convergence of Kleinian groups, "[A] sequence $\{G_n\}$ of Kleinian groups converges geometrically to a Kleinian ...
- 61
1
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102
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Are there four dimensional generalizations of the Reuleaux triangle and other solids of constant width? [closed]
Is there a four dimensional generalization of the Reuleaux triangle? What is it called, and what properties does it have?
Thank you!
- 119
4
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275
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Billiard circuits in pentagons
A billiard circuit in a convex $n$-gon is a closed billiard path
of $n$ segments reflecting from consecutive edges of the polygon.
Every regular $n$-gon has such a billiard circuit:
Recently a ...
- 149k
5
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1
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124
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Expected value of the length of the shortest non-zero vector in a lattice?
$\DeclareMathOperator\SL{SL}$What is the expected value of the length of the shortest non-zero vector in a (unimodular) lattice? I.e., let $G=\SL_n(\mathbb{R})$ with Haar measure $\mu$, $\Gamma=\SL_n(...
- 507
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157
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Root system terminology
Let $\Phi$ be a root system. In a paper I'm writing, I need to work with subsets $\Phi' \subset \Phi$ satisfying the following two conditions:
For all $\lambda_1,\lambda_2 \in \Phi'$ and $c_1,c_2 \...
- 21
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676
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Where to find English translation of Pansu's paper from Ann. Math?
Where can I find English translation of the following paper?
P. Pansu,
Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. (French. English summary) [Carnot-...
- 27.3k
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Unusual parameterization of the ring Dupin cyclide
I discovered the following by playing with the formulas given in the paper Sculptures in $S^3$ by Schleimer and Segerman.
First, define the following parameterization of the Clifford torus:
$$
p(\...
- 2,433
5
votes
1
answer
88
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How to bulge out the curved edges of the stereographic tesseract?
You probably already saw such a representation of the tesseract:
I did something similar on my blog for the truncated tesseract:
The vertices in 3D are the stereographic projections of the original ...
- 2,433
4
votes
1
answer
347
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Upper bounds on the Gromov–Hausdorff distance using persistent homology
In persistent homology theory, stability theorems are important to show that the topological signatures extracted are stable under small changes. A key result is the following bound on the bottleneck ...
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3
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931
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Is there a pyramid with all four faces being right triangles? [closed]
If such a pyramid exists, could someone provide the coordinates of its vertices?
3
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143
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Algorithm to dissect a polygon into a minimum amount of rectangles, conditioned on a maximum overlap
I have the following problem, I have a problem regarding concave polygons. I want to write code to cover any polygon with a minimum amount of rectangles that are allowed to overlap and have no fixed ...
- 31
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33
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Minimum diameter of spherically-inverted topological balls
Let $U$ be the closed unit ball in $\mathbb{R}^3$. Let $S$ be a round sphere whose center is in $U$ with radius at least $\delta_1 > 0$. Suppose $B$ is a closed topological ball of Euclidean ...
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Does the Alexandrov angle define convex functions along geodesics in CAT(0) spaces?
Let $X$ be a CAT(0) space and suppose $a,b,c,d\in X$ satisfy
$$
\max\{\angle_a(b,c),\angle_a(b,d)\}<\frac{\pi}{2}.
$$
Let $\gamma:[0,\ell]\to X$ be the geodesic with $\gamma(0)=c$ and $\gamma(\ell)...
- 741
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81
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relative entropy, Fisher information, and metric slope for non-convex domains
$\newcommand{\R}{\mathbb R}$ If $\Omega\subset \R^d$ is a convex domain it is well-known that the relative entropy
$$
\mathcal H(\rho)=
\int_{\Omega}\rho\log\rho \ \mathrm{d}x
\qquad \mbox{for }\rho=...
- 5,163
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106
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Is there an inventory of closed billiard paths in a regular tetrahedron?
Conway found a closed billiard-ball trajectory in a regular tetrahedron:
Image: Izidor Hafner
Since then Bedaride and Rao
Bedaride, Nicolas, and Michael Rao. "Regular simplices and periodic ...
- 149k
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0
answers
84
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Number of polyhedral covers of a triangulation of $S^2$
For a given triangulation (combinatorial Type I. or Type II.) of a $2$-sphere, what is the number of unique polygonal covers with $n$ polygons where ($n$ goes from $2$ to $N$)?
Under polygonal cover, ...
- 75
1
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104
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Do Gromov hyperbolic spaces admit concical geodesic bicombings?
Consider a metric space $(X,d)$ with a distinguished selection of geodesics, i.e. a geodesic bicombing $\sigma:X\times X\times [0,1]\rightarrow X$. We call a geodesic bicombing conical if it ...
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1
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357
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Illumination from visible lattice points with inverse square intensity
It is well known that the number of $\mathbb{Z}^2$ lattice points visible from
the origin is $6/\pi^2$, about $61$%.
See, e.g.,
What fraction of the integer lattice can be seen from the origin?.
I am ...
- 149k
7
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234
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How did Szmielew prove that Pasch's axiom is a consequence of the circle axiom?
It is alleged that Szmielew proved that Pasch's axiom is a consequence of the circle axiom. The source is said to be
The Pasch axiom as a consequence of the circle axiom, Bull.Acad.Polon.Sci.Sér.Sci....
11
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0
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661
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John-type theorems: trading structure for accuracy?
Given two symmetric convex bodies $B, B'$ in ${\bf R}^d$, define the Banach-Mazur distance $d(B,B')$ between them to be the least constant $\tau \geq 1$ such that
$$ B \subset TB' \subset \tau B$$
for ...
- 109k
2
votes
1
answer
122
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Is the vector field associated with an element of the boundary at infinity on a Hadamard manifold smooth?
A Hadamard manifold $M$ (complete, simply connected, non-positive sectional curvature) has a so-called boundary at infinity $\partial M$ whose elements are equivalence classes of unit-speed geodesic ...
- 23
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776
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Is this set a manifold?
Take a general spacetime that is not strongly causal.
Call this spacetime $(M, g) $ where $M$ is a connected time-oriented manifold and $g$ is the Lorentzian metric that satisfies the Einstein's Field ...
- 308
2
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0
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45
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Estimating the Hausdorff distance of parallel facets of convex polytopes
Background
Let $\mathcal{K}_P^n$ denote the class of open, convex, $n$-dimensional polytopes in $\mathbb{R}^n$ containing the origin. For each $K\in \mathcal{K}_P^n$, its gauge function $f_*:\mathbb{R}...
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