Questions tagged [mg.metric-geometry]

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

Filter by
Sorted by
Tagged with
2 votes
0 answers
70 views

Converse of existence of minimizers

Let $(V,\|\cdot\|)$ be a real normed linear space. $V$ has the property that given any nonempty convex, closed subset $K$, there exists a unique $v_0\in K$ such that $\|v_0\| \leq \|v\|, \forall v\in ...
Rohan Didmishe's user avatar
3 votes
2 answers
757 views

Kepler conjecture: Are there only two most efficient packings or could there be more than two?

Today I attended a talk by Terence Tao, attended by (I'm guessing) probably at least a couple of thousand people, in which among other things he said it had been proved that no packing of spheres in ...
Michael Hardy's user avatar
2 votes
1 answer
96 views

Another lemma on intersections of $d$-simplices

Let $d\ge1$. A $d$-simplex $S$ is the convex hull in $\mathbb R^d$ of the vertices $v_0,\dots,v_d\in\mathbb R^d$ where $\{v_1-v_0,\dots,v_d-v_0\}$ is a linearly independent set of $d$ vectors; for ...
Tri's user avatar
  • 1,366
0 votes
0 answers
35 views

Determining a convex hyperbolic pentagon by all side lengths and two specified angle sums

We are trying to prove the following statement for convex hyperbolic pentagons which we believe should be true. Consider a convex hyperbolic pentagon with sides of lengths $a, b, c, d, e$. Suppose the ...
Shiv Parsad's user avatar
1 vote
0 answers
44 views

Computational complexity of exact computation of the doubling dimension

Given a finite metric space $X$, the doubling constant of $X$ is the smallest integer $k$ such that any ball of arbitrary radius $r$ can be covered by at most $k$ balls of radius $r/2$. The doubling ...
pyridoxal_trigeminus's user avatar
0 votes
1 answer
108 views

How to estimate the distance between geodesics and points for Riemannian manifold with positive sectional curvature

Assume that $ M $ is a complete Riemannian manifold and there exists $ k>0 $ such that $ K(q)\geq k $ for any $ q\in M $, where $ K $ is the sectional curvature of $ M $. Let $ \gamma $ be a closed ...
Luis Yanka Annalisc's user avatar
1 vote
1 answer
305 views

Geometry in $\mathbb{R}^n$: angle between projections of a rectangle

Consider a hyper rectangle $R$ in $\mathbb{R}^n$ defined by $|x_i|\leq M_i$ for all $i\leq n$. Consider a linear affine subspace $L$ of dimension $1\leq k <n$ such that $L\cap R\neq \emptyset$. For ...
Alainty's user avatar
  • 11
1 vote
0 answers
41 views

Intersection of unit-norm vectors with a large sum in high dimensions with a spherical cap

Let $d$ and $n$ be integers. For $i \in \lbrace 1,\dots,n \rbrace$ let $x_i \in \mathbb{R}^d$ be a vector such that $\lVert x \rVert=1 $. For a fixed $1/2 < \alpha \leq 1$, assume we have $\lVert \...
MMH's user avatar
  • 129
11 votes
0 answers
479 views

Are there 100 points that are part of every half-density part of the plane?

Is there a configuration $P$ that consists of 100 points of the plane such that every $X\subset\mathbb R^2$ whose density is half contains an isometric copy of $P$? I am deliberately being vague ...
domotorp's user avatar
  • 18.3k
2 votes
0 answers
136 views

$\mathscr{H}^{n-2}(\Sigma)< \infty$ implies $\mathscr{H}^{n-1}(\pi(\Sigma))=0$

Let $\Sigma\subset \mathbb{R}^{n+1}$ be a set with $(n-2)$-dimensional Hausdorff measure finite, i.e. $\mathscr{H}^{n-2}(\Sigma)<\infty$. Let $\pi:\mathbb{R}^{n+1}\to \mathbb{R}^n$ be the ...
No-one's user avatar
  • 1,037
0 votes
0 answers
29 views

Minimizing the number of grid squares to cover a polygon

Given an arbitrary polygon, and a grid square size x, I'd want to find a placement of the polygon such that it covers the minimum amount of cells in the grid. The ...
b9s's user avatar
  • 101
0 votes
1 answer
54 views

Robustness of doubling dimension to small perturbations

Let $M$ be a metric space. Then the doubling dimension of $M$, denoted $\dim M$, is defined to be the minimum value $k$ such that every ball in $M$ of radius $r$ can be covered by at most $2^k$ balls ...
pyridoxal_trigeminus's user avatar
0 votes
1 answer
76 views

Kähler metric on the projective space

"Is there a Kähler metric on the complex projective space $\mathbb {P} ^n(\mathbb {C} ) $ different from the Fubini-Study metric?
Samir's user avatar
  • 43
3 votes
0 answers
49 views

Volume of all Voronoi cells in n-dimensional bounded space

How can one find the volume of all Voronoi cells (bounded and unbounded) in an $n$-dimensional bounded space? For instance, consider an $N$-dimensional space (hypercube) with bounds on each dimension ...
Maaz's user avatar
  • 131
1 vote
0 answers
146 views

Geometric construction of real root of quintic using marked ruler and compass

My question is motivated by a geometry problem about special folded rectangle: 'A rectangle with sides a, b (a<b) is folded along the line that passes through the center of the rectangle in order ...
Mikhail Gaichenkov's user avatar
1 vote
1 answer
110 views

In the limit, do the intersection points of a string figure define a probability measure on the unit disk?

Let D = {z ∈ ℂ | |z| ≤ 1} denote the closed unit disk in the complex plane. For any integer n ≥ 1 define the nth string figure S(n) ⊂ D as the union of all n(n+1)/2 line segments that connect two ...
Daniel Asimov's user avatar
0 votes
0 answers
121 views

Naming convention for different type of triangulations

When studying random geometries and related mathematical/physical stuff conflicting naming convention pops up regarding the naming of the different ensemble types of triangulations (in general ...
Kregnach's user avatar
0 votes
0 answers
27 views

Is there a bi-Hölder Weierstrass-type embedding of the circle into some Euclidean space?

We say that $\Phi\colon S^1\to \mathbb{R}^d$ is an $\alpha$-bi-Hölder embedding if there are constants $c_1,c_2>0$, such that $$c_1\leq \frac{\|\Phi(x)-\Phi(y)\|}{d(x,y)^\alpha}\leq c_2,$$ where $d$...
Roope Anttila's user avatar
1 vote
0 answers
104 views

Which polygons allow partition into rational triangles?

A triangle with all side lengths rational is said to be a rational triangle. It is known - for example, Cutting the unit square into pieces with rational length sides - that the unit square allows ...
Nandakumar R's user avatar
  • 5,401
1 vote
1 answer
116 views

Divergence functions in hyperbolic groups

Gromov hyperbolicity has many characterizations, one of them being the existence of a super-linear divergence function, see definition below. We note that in $\mathbb{R}^2$ there is no divergence ...
Strichcoder's user avatar
3 votes
0 answers
122 views

Conformal Killing vector fields on manifolds that are not asymptotically flat

Let $M = [1,\infty) \times S^2$. Equip $M$ with the metric $g = dr^2 + r^2 (\gamma + h)$ where $\gamma$ is a metric on $S^2$ and $h$ is a $(0,2)$ tensor on $M$ that satisfies $$h = O(1/r),\quad \...
Laithy's user avatar
  • 865
0 votes
2 answers
95 views

Conditions on a parametric curve so that its normal plane covers R^n

I am working on a control theory problem that either just caught me on a blind spot or is beyond me. I guess it's not a new question, but I couldn't find anything about it. Let $p(s), s\in \mathbb{R}, ...
user517853's user avatar
5 votes
1 answer
122 views

Variants of the Bonk-Schramm embedding

Recently I heard about the following embedding theorem of Bonk and Schramm: every Gromov hyperbolic geodesic metric space with "bounded growth" is roughly similar to a convex subset of $\...
Takao Hishikori's user avatar
3 votes
1 answer
151 views

What is the minimum and the maximum perimeter of a triangle with area $x$ that can be inscribed in a circle?

This question was posted in MSE but is still open hence posting in MO. The area of the largest triangle that can be inscribed in a circle of raidus $1$ is $\displaystyle \frac{3 \sqrt{3}}{4}$ for a ...
Nilotpal Kanti Sinha's user avatar
4 votes
4 answers
283 views

Groups acting non-properly cocompactly on hyperbolic spaces

A group $G$ is hyperbolic if it admits a geometric (the action is proper and co-bounded) action on a geodesic hyperbolic metric space. Also, the definition can be given as follows, a group $G$ ...
bishop1989's user avatar
1 vote
1 answer
101 views

Fitting a simplex to set of almost orthogonal vectors

I am trying to solve the following question, that I guess (hope?) has been solved before but I couldn't find any reference. Let $S$ be a set of $d$ unit vectors in a $d$-dimensional Euclidean space ...
user103464's user avatar
3 votes
1 answer
150 views

Injective hulls of metric spaces

In the context of large scale geometry and geometric group theory, I have recently come across the concept of injective hulls of metric spaces. For a metric space $X$, let $\text{In}(X)$ be the set of ...
Sebastian's user avatar
20 votes
1 answer
902 views

Conjecture: Given any five points, we can always draw a pair of non-intersecting circles whose diameter endpoints are four of those points

The following question resisted attacks at Math SE, so I thought I would try posting it here. Is the following conjecture true or false: Given any five coplanar points, we can always draw at least ...
Dan's user avatar
  • 2,341
1 vote
0 answers
76 views

Moser iteration epsilon-regularity for non-linear system in general dimension

I am attempting to prove the following result in general dimension $n$. Given $(M^n,g)$ a Riemannian manifold with $\mathrm{Ric}_g \geq -(n-1)$ and $\mathrm{Vol}_g(B_1(x)) \geq v > 0$ for all $x \...
Curious DeGiorgio's user avatar
1 vote
1 answer
173 views

Average distance between points of lower dimensional simplices in $\mathbb R^n$

Notation: By a simplex, we mean the convex hull of a finite set of distinct points in $\mathbb R^n$, which are called the vertices of the simplex. $\mathcal H^n$ will denote the $n$-dimensional ...
Nate River's user avatar
  • 4,802
3 votes
0 answers
54 views

A question about the existence of surjective contractions

A few years ago I was doing some research in origami, and was motivated to as the following questions: Consider $\mathbb{R}^2$ with the Euclidean metric and Lebesgue measure. Does there exist a ...
abacaba's user avatar
  • 344
0 votes
1 answer
33 views

L2 distance computation with given distance to triangle nodes [closed]

In a triangle with three points A, B, and C. The L2 distance between each pair of points |AB|, |AC|, |BC| is given. For the other two points O and P, the distance to the three points is given, i.e. |...
Yujian S's user avatar
1 vote
1 answer
134 views

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\...
Learning math's user avatar
4 votes
1 answer
163 views

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 ...
Mohammad Ghomi's user avatar
4 votes
2 answers
235 views

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 ...
theflame's user avatar
2 votes
1 answer
103 views

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/...
Nandakumar R's user avatar
  • 5,401
14 votes
0 answers
367 views

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 ...
Dan's user avatar
  • 2,341
3 votes
0 answers
96 views

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 $k$ ...
Steve Riley's user avatar
4 votes
0 answers
157 views

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 ...
Dan's user avatar
  • 2,341
7 votes
2 answers
436 views

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 $...
bishop1989's user avatar
1 vote
0 answers
102 views

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^...
ABIM's user avatar
  • 4,989
22 votes
1 answer
3k views

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. ...
Nate River's user avatar
  • 4,802
11 votes
1 answer
387 views

Smallest sphere containing three tetrahedra?

What is the smallest possible radius of a sphere which contains 3 identical plastic tetrahedra with side length 1?
trionyx's user avatar
  • 111
2 votes
1 answer
108 views

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 ...
Stephen Jiang's user avatar
2 votes
0 answers
149 views

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 ...
Enhao Lan's user avatar
  • 165
0 votes
0 answers
172 views

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 ...
Claudio's user avatar
1 vote
0 answers
101 views

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 ...
Mohammad Ghomi's user avatar
3 votes
1 answer
377 views

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 ...
Nate River's user avatar
  • 4,802
2 votes
1 answer
264 views

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 ...
No-one's user avatar
  • 1,037
2 votes
0 answers
128 views

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 ...
jsch's user avatar
  • 21

1
2
3 4 5
85