Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,406 questions
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On packing axisymmetric bodies in 3D
Consider any 3D body with an axis of rotational symmetry (e.g. cone, cylinder...) and packing the 3d space efficiently with infinitely many copies of this body. Is the following claim valid?
Claim: ...
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Pythagorean triples related to non-isometric equidistant plane quadruples
QUESTION Do there exist integers $u\ x\ A\ B$ such that $x\ne 0$, and the following two equalities hold:
$ x^2 + (x-u)^2\ =\ A^2$
$ x^2 + (x+u)^2\ =\ B^2$
?
...
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509
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A question concerning a well known "law" about triangles.
Let a,b,c denote the lengths of the sides and let A,B,C denote the corresponding opposite angles of
a triangle. In the Euclidean plane we have the law of sines. a/sin(A)=b/sin(B)=c/sin(C). A recent
...
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Can a set of tetrahedra glued together by a common vertex be isometrically embedded in R^4?
A collection of triangles with a common vertex $A_1VA_2$, $A_2VA_3$, ... $A_NVA_1$ with specified side lengths can be isometrically embedded in $R^2$ provided the angles around $V$ add up to $2\pi$. ...
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Antiproximanal subspace of $L_1[0,1]$
Could someone give a reference or construct an example of closed subspace of $Y\subset L_1[0,1]$ such that $\operatorname{dist}(x,Y)$ is not attained of for any $x\notin Y$.
I read somewhere that $Y$ ...
- 1,697
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Isometric imbedding of finite metric space into standards spaces [duplicate]
Is it true that any metric space consisting of $n$ points can be isometrically imbedded into $n-1$ dimensional Euclidean space? Hyperbolic space?
(For $n=3$ this is true.) If not, what are necessary/...
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Determining the maximum number of distance relationships that can be defined between points in Euclidean space
Let there be $m$ points in the Euclidean space $\mathbb R^n$. Randomly choose $k$ distinct pairs of these $m$ points, and assign a random (positive) value for the Euclidean distance between each of ...
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Non-compact surfaces with non-negative Gauss curvature
Is there a topological classification of non-compact complete connected 2-dimensional Riemannian manifolds with non-negative Gauss curvature?
- 21.8k
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What is the symmetry group of this compound of two polytopes?
The geometric shape in question is a compound of two polytopes: an 11-hypercube with edge length $2$ and an 11-simplex with edge length $\sqrt6$ whose vertices are a subset of the hypercube’s. What is ...
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What information about the lattice $\Lambda$ can be recovered from the metric space $\mathbb{R}^n/\Lambda$?
Let $\Lambda\subset\mathbb{R}^n$ a lattice, i.e., a discrete subgroup that spans $\mathbb{R}^n$. Now we can look at the torus $T=\mathbb{R}^n/\Lambda$ which naturally carries the metric $d_T$ induced ...
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Strictly increasing functions in reflexive subspaces of $C([0,1])$
By the Banach-Mazur theorem, every separable Banach space $X$ embeds into $C([0,1])$. When $X$ is reflexive, it is not possible to find a sequence of disjointly supported, non-negative functions in ...
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Prospects for deep learning of non-lattice sphere packings
I have been looking for litterature on results obtained by deep neural networks to find dense (and quite possibly non-lattice, perhaps even non-periodic) sphere packings, but I have not been too ...
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What’s the relation between Lehmer’s Conjecture and Systole [closed]
What’s the relation between Lehmer's Conjecture and Systole?
Lehmer’s conjecture says: There exists $m>1$ such that $M(p)\geq m$ for all noncyclotomic $P$.
Systole is a closed geodesic of the ...
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Sectional curvature as a Hamiltonian on the Grassmanization of the tangent bundle
Edit: According to the comments to the previous version of this question, I remove my essential errors in the question. I thank the commenters very much.
Let $M$ be a n dimensional manifold. ...
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A question about metric spaces that are compact and denumerably infinite
Let S be any compact and denumerably infinite metric space and let d be the metric of S. We shall say that S satisfies condition C, if there exists at least one infinite sequence-------------p(1),p(2),...
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a question about $\epsilon$ net of a compact metric space.
A subset A of a compact metric M is called a $\epsilon$ net if it satisfies the following conditions
(1)$\epsilon$ dense: the neighborhood of A is the entire M
(2)$\epsilon$ separate: $\forall x, y ...
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The probability a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice takes more than $N$ steps before trapping itself
What is the probability that a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice is able to take more than $N$ steps, i.e. able to take more than $N$ steps before trapping itself ...
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triangle equality in manifold
For a generilized triangle on a manifold, (distance can be regarded as geodesic length)it is well known that for Eucilidean Geometry，the following is true:
Consider a triangle $ABC$, $D$ is the ...
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Is there a good approximating polygon for every smooth set?
Suppose we have an open set S whose boundary is a closed Jordan curve that has a unique tangent at each point. Is it true that for every epsilon there is a P polygon contained in S such that there is ...
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Does $\mathbb Z^n$ contain $A_n$?
Are there any positive integer $n > 3$ such that the root lattice $A_n$ is contained in $\mathbb Z^n$?
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A triangle comparison in CAT(0) spaces
Let $pxy$ be a triangle in a CAT(0) space $X$, and $p' x' y'$ be a triangle in $\mathbf{R}^2$ such that the lengths $|px|=|p'x'|$, $|py|=|p'y'|$ and the angle $\angle(xpy)=\angle(x'p'y')$. Let $z\in ...
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Collinearity in tangential pentagon [closed]
I am looking for a proof of the following claim:
Given tangential pentagon. Touching point of the incircle and the side of the pentagon,the vertex opposite to that side and the intersection point of ...
- 2,661
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Approximate the following series on the euclidean grid
I had trouble in finding a closed form solution for the following series, so now I am trying to find a good approximation for it. The $\sqrt{i^2 + j^2}$ in the exponent comes from distances on the ...
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714
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State-of-the-art geometry book? [closed]
For my best friend's birthday, I am looking for a geometry book. He's currently doing his math PhD and is really fond of geometry, especially hyperbolic or higher-dimensional ones, also interested in (...
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Extreme points and centroid
It is well-known that the centroid of a triangle is the intersection point of its three medians. The medians happen to be area bisectors, but it seems that most (all?) other lines through the ...
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Hales's fan associated with a polyhedron
In Hales's book (cited below), he associates what he calls a fan with any convex polyhedron in $\mathbb{R}^3$.
I will not define his notion of fan, but let his figure (p.137) serve as a definition:
...
- 151k
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A question concerning convex functions
Consider points $a,b,c$ (not on a line) and $x_1,...,x_n$ in $\Bbb{R}^2$. I am looking for a necessary and sufficient condition in terms of the geometric configuration of these points such that for ...
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the tangent cone at infinity of manifolds with Rc\geq 0
Is the following statement true?
Assume that $(M^n, g, p)$ is a pointed complete manifold with metric $g$, $Rc(g)\geq 0$,
$\{s_i\}$ is a positive sequence decreasing to $0$, and $(M^n, s_i\cdot g, p)$...
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sobolev embedding theorem in the smooth metric measure space
we know the sobolev embedding theorem of Saloff-Coste
$\Big(\int_B|F|^{2q}d\mu\Big)^{\frac1q}\le e^{C(1+\sqrt KR)}V^{-2/n}R^2\int_B\Big(|\nabla F|^2+R^{-2}F^2\Big)d\mu
$
wtih $Ric\ge-(n-1)K$, for ...
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A question about dissecting spherical triangles
Do there exist spherical triangles which are not isoceles but are the union of a finite collection of
(two or more) congruent triangles with pairwise disjoint (and non-empty) interiors?
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How to prove that the hyperbolic space is $\delta$-hyperbolic
How can I prove that the hyperbolic space $\mathbb{H}^n$ (in any of its realization as hyperboloid, Poincaré disc or Poincaré half-space) is $\delta$-hyperbolic? For the moment I am not interested in ...
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Who was Harry Hart?
It appears that some geometers who know about Hart's Theorem are not familiar with Hart's Inversors, and conversely. Can someone locate a biographical sketch of Harry Hart?
As an aside, I must say ...
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Probability that three vectors of a unit sphere lie on one side of a hyperplane if angle between the vectors are given
As the title says, How to find the probability of vectors a, b, c, on some unit sphere, all lies on same side of some hyperplane passing through the origin. Information present are the angles between ...
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On convex planar regions that can be cut into only a specified number of mutually congruent and connected pieces
References:
https://math.stackexchange.com/questions/1838617/dividing-an-equilateral-triangle-into-n-equal-possibly-non-connected-parts
On congruent partitions of planar regions
https://research....
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A question about dense sets
Suppose that $A$ is a given subset of $I=[0,1],\ $ and
$ \left\{ x_j = \frac{j}{m} \right\}_{j=0}^{m}\ $ is the $m$-partition of $I$, and $\nu(m)$ is the number of
$\ [x_{i-1},x_{i}]\ $ such that $\ [...
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Doubling constant of Carnot group
This post shows that every Carnot group is a doubling metric space. However, what is its doubling constant?
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Broken geodesic in Finsler polyhedral space
Here we assume that all norms has only one geodesic, i.e. locally
minimizing, between any two points.
Example : In $\mathbb{R}^2$, a line $y=kx,\ k>0$ divides
$\mathbb{R}^2$ into two regions. We ...
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A possible characterization of the cube?
Let $P$ be the $1$-skeleton of a convex polyhedron fixed in $\mathbb{R}^3$,
and $|P|$ the sum of the Euclidean lengths of the edges of $P$.
Let $P_1, P_2, P_3$ be the perpendicular projections of $P$
...
- 151k
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A possible characterization of Euclidean geometry via the curvature of the Median-submanifold
Is there a Riemannian metric $g$ on $\mathbb{R}^2$ with inducing distance
$d$ which is not isometric to the standard metric but satisfy the property quoted bellow?
For every two distinct ...
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Minimizing maximum distance by adding shortcuts in grid graph
My problem is to find places to put k number of shortcut edges with weight 0 to minimize maximum distance in grid graph where all edges are weighted 1!
I found a related topic to my question ...
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Is there a law of cosine for n-dimensional hyperbolic simplex
We know that, given an n-dimensional Euclidean simplex, for all $1\leq i,j,k,l\leq n+1$, we have(law of sines)$$\frac{A_i A_j}{A_k A_l}=\frac{c_{ij}}{c_{kl}}$$(from Elementary Formulas for a ...
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Name of area between two parallel lines [closed]
Assume that there are two distinct parallel lines on a Euclidean plane. Is there a name for the zone between these two lines?
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can every Gromov-Hausdorff limit be realized as a metric ultralimit?
Let $(X_i)$ be a countable collection of bounded metric spaces, pre-compact in the Gromov-Hausdorff metric. It is well-known that for any choice of non-principal ultrafilter $U$ on $\mathbb N$, the ...
- 4,070
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Alexandrov spaces of constant curvature
Let $X$ be locally compact Alexandrov space whose curvature satisfies both inequalities $\geq K$ and $\leq K$. What can be said about such a space? Is it locally isometric to the standard Riemannian ...
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Optimal fitting of spheres in a cylinder
How to find the minimum height and width of a cylinder containing n identical spheres?
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How do maximum norms relatively change in Euclidean translations
Let $Q$ be the cube $[-1,1]^{3}$ and $\pi$ be a plane in $\mathbb{R}^{3}$
that contains the origin but doesn't contain any vertex of $Q$. Suppose that $A$ is an invertible
linear transformation from $\...
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Is there always a parallelogram cross-section of parallelepiped contained in the smallest box
Let $M$ be a centered parallelepiped, the intersection of $M$ and any plane $P$ that passes through the origin is a parallelogram or hexagon. Each parallelogram or hexagon has a cubic box that is the ...
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Metric currents on singular measures in $\mathbb R^d$
Unless I am misunderstanding a lot of works, it is my understanding that a finite and non negative measure $\mu=g\mathcal{H}^\alpha$, where $\mathcal{H}^\alpha$ is the $\alpha$-Haudorff measure, ...
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Relation between the root lattice of $\mathrm{SO}(7)$ and the root lattice of $G_2$
The root lattice of $\mathfrak{so}(7)$ is given by the following 18 roots:
$$
\left(\begin{array}{c}0\\0\\1\end{array}\right)
,
\left(\begin{array}{c}0\\0\\-1\end{array}\right)
,
\left(...
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