Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,249
questions
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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 ...
3
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2
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757
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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 ...
2
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1
answer
96
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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 ...
0
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35
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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 ...
1
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0
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44
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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 ...
0
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1
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108
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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 ...
1
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1
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305
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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 ...
1
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0
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41
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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 \...
11
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479
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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 ...
2
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136
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$\mathscr{H}^{n-2}(\Sigma)< \infty$ 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 ...
0
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29
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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 ...
0
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1
answer
54
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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 ...
0
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1
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76
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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?
3
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49
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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 ...
1
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0
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146
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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 ...
1
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1
answer
110
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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 ...
0
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121
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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 ...
0
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0
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27
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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 $d$...
1
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0
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104
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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 ...
1
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1
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116
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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 ...
3
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0
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122
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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 \...
0
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2
answers
95
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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}, ...
5
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1
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122
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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 $\...
3
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1
answer
151
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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 ...
4
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4
answers
283
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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$ ...
1
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1
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101
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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 ...
3
votes
1
answer
150
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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 ...
20
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1
answer
902
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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 ...
1
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0
answers
76
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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 \...
1
vote
1
answer
173
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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 ...
3
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0
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54
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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 ...
0
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1
answer
33
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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. |...
1
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1
answer
134
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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\...
4
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1
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163
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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 ...
4
votes
2
answers
235
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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 ...
2
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1
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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/...
14
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0
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367
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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 ...
3
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0
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96
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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$ ...
4
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0
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157
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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 ...
7
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2
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436
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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 $...
1
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0
answers
102
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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^...
22
votes
1
answer
3k
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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. ...
11
votes
1
answer
387
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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?
2
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1
answer
108
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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 ...
2
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0
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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 ...
0
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0
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172
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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 ...
1
vote
0
answers
101
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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 ...
3
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1
answer
377
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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 ...
2
votes
1
answer
264
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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 ...
2
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0
answers
128
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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 ...