Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,405 questions
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Non-Kahler Complex manifolds
For a non-Kahler complex manifold $M$, we still have the decomposition of differential forms into differential forms of type $(p,q)$ and we can write $d=\partial+\bar\partial$ and we can define ...
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Which curves and surfaces are realizable by linkages? references?
Ok, so I try to formulate rigorously the question in the title, for which I am asking for references. My definitions may be flawed, so feel free to adjust/correct them! I care about dimensions 2 and 3 ...
- 2,041
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Characterization of Riemannian metrics
This is probably an insanely hard question, but given an abstract metric space, is there some way to determine whether it's a manifold with a Riemannian, or more generally a Finslerian, metric? If ...
- 1,665
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Place of Analytic geometry in modern undergraduate curriculum
I am a freshmen student in mathematics at Moscow State University (in Russia) and I'm confused with placing the subject called "analytic geometry" into the system of mathematical knowledge (if you ...
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Sequential addition of points on a circle, optimizing asymptotic packing radius
Suppose I have to put $N$ points $x_1, x_2, \ldots, x_N$ on the circle $S^1$ of length 1 so as to achieve the largest minimum separation (packing radius). The optimal solution is the equally spaced ...
- 5,971
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Limit of distance between two random points in a unit $n$-cube
What is the limit, as $n \to \infty$, of the expected distance between two
points chosen uniformly at random within a unit edge-length hypercube
in $\mathbb{R}^n$?
For $n=1$, the average distance ...
- 151k
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Sums of squared distances between points on an $n$-sphere
I have discovered the following results about the sums of squared distances between points on an $n$-sphere (and proved them). To the best of my knowledge (and my advisor's knowledge), these results ...
- 155
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Continuity of barycentre in Hausdorff metric
Let $K_1$, $K_2$ be two convex compact sets in $\mathbb{R}^d$, and $p_1,p_2$ be their barycenters. Is it true that the distance between $p_1$ and $p_2$ does not exceed a Hausdorff distance between $...
- 109k
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Eigenvalues of Laplacian-Beltrami operator
I am interested in the first non zero eigenvalue of the Laplace-Beltrami operator in a 2D compact manifold, and if there is a geometric characterization of its value.
I am interested in the case when ...
- 163
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Decomposing the plane into intervals
I posted this on Stack Exchange and got a lot of interest, but no answer.
A recent Missouri State problem stated that it is easy to decompose the plane into half-open intervals and asked us to do so ...
- 356
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Dodecahedral rolling distance
Let a dodecahedron sit on the plane,
with one face's vertices on an origin-centered unit circle.
Fix the orientation so that the edge whose indices are $(1,2)$ is horizontal.
For any $p \in \mathbb{R}...
- 151k
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What is the area of the biggest open convex set inside the unit square not containing k points?
Given $k\in \mathbb N$, and $k$ points inside the unit square there should be an arrangement that minimizes the area of the biggest open convex set inside the unit square not containing these points.
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Mathematics of doodling and the winding number
So I was reading the American Mathematical Monthly Feb 2011 (Volume 118, number 2), and in particular, I was interested in Ravi Vakil's article about mathematics of doodling. There is a question I ...
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Geodesics on SO(3)
I have two 3D rotations about the origin, represented as
$3 \times 3$ orthogonal matrices $M_1$ and $M_2$
(specified by numerical entries),
and I would like to interpolate (and compute)
a continuous ...
- 151k
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Algebraicity of the completion of a field? Finiteness?
At the end of my 8410 class today (see http://alpha.math.uga.edu/~pete/MATH8410.html if you care), one of my students asked me the following very interesting question:
Let $(K,|\ |)$ be a normed field,...
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What is the minimum-curvature curve interpolating a given set of points in the plane?
We are given a set $X$ of $n\ge 3$ points in $\mathbb{R}^2$, belonging to the boundary of the convex hull of $X$ itself. Let $\Gamma(X)$ be the set of all convex, simple closed curves in $\mathbb{R}^2$...
- 1,677
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How to construct a nice homotopy?
Let $(M,g)$ be a closed simply-connected Riemannian manifold. Can we find a constant $C$, which depends on $M$, such that for any closed curve $\alpha_0$ with $L(\alpha_0) \le 1$, there exists a ...
- 2,535
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Quasi-isometric groups without common virtual geometric model
Are there known examples of finitely generated groups $G$ and $H$ that are quasi-isometric but do not admit finite-index subgroups $G'<G$ and $H'<H$ such that both $G'$ and $H'$ admit proper and ...
- 2,050
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Five points in spheres
Do there exist five points in the euclidean space ${\mathbb R}^3$ such that
every four of these points are in a spherical ball of radius 1, but that the five points are not in a ball of radius 1?
Do ...
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polyhedra with equilateral pentagons faces
In page http://loki3.com/poly/isohedra.html around six polyhedra with equilateral pentagons as faces are shown: a pyritohedron, icositetrahedrons... Is there a complete list of this kind of polyhedra? ...
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Geometry of the multilagrangian Grassmannian
Let's introduce the following variety $MG(3,6)$, which is a "multisymplectic" analog of a Lagrangian Grassmannian $LG(3,6)$.
Consider a 3-form $\omega = dx1 \wedge dx2 \wedge dx^3 - dx4 \wedge dx5 \...
- 1,783
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Book on the tetrahedron
Does anybody know of a book containing "all you want to know about the tetrahedron"? What you want to know should include basic geometry of the tetrahedron, study of orthocentric tetrahedra, the Monge ...
- 4,492
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Floating polyhedra with fair equilibria
Is there a homogeneous convex polyhedron
which floats so that some subset (perhaps all) of its faces
is distinguished as "up" (above the water line)
in stable equilibrium, each face with equal ...
- 151k
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Strange formula for area of a convex polygon
Consider a convex $n-$gon in $\mathbb{R}^2$ with sides contained in the lines $y=k_ix+b_i, 1\leq i\leq n.$ Then its area equals to
$$
S=\frac{1}{2}\sum_{i=1}^{n} \frac{(b_{i+1}-b_i)^2}{k_{i+1}-k_i}.
$$...
- 4,316
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Source for: Geodesics in CAT(0) spaces
I am seeking a good introductory reference that could lead to an understanding of
the properties of geodesics in
complete CAT(0) metric spaces.
I am especially interested in learning the differences ...
- 151k
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Riemannian metrics as sections of a vector bundle
Let $\pi \colon E \to M$ be a smooth vector bundle. A Riemannian metric on $E$ can be regarded as a global section of the vector bundle $(E\otimes E)^{\ast}$, or more specifically, the subbundle $(S^...
- 19.3k
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Is Stoch enriched in Met?
Let $\mathsf{Stoch}$ denote the Kleisli category of the Giry monad. That is, $\mathsf{Stoch}$ is a category whose objects are measurable spaces and for which a morphism $f\in\mathsf{Stoch}(X,Y)$ is a ...
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Thinnest 2-fold coverings of the plane by congruent convex shapes
It is an unsolved problem to determine the "thinnest" $2$-fold covering of
the plane by disks.
The $2$-fold coverage problem by disks is to find the minimum number of congruent
(unit-radius) disks ...
- 151k
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"Rolling Geodesics": Designing a $k$-putt green
I am interested in what might be called rolling geodesics, paths
of physical particles confined to a surface in $\mathbb{R}^3$
under certain force conditions.
Here I will pose a specific (but ...
- 151k
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Isometric (?) embedding problem.
Given a convex surface $S$ in $\mathbb{R}^3,$ you can associate to each point the length of the inward pointing normal contained inside $S.$ Call this quantity $s(x).$ Now, given a positive function $...
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How many non-equivalent sections of a regular 7-simplex?
Suppose we have a regular 7-simplex in $\mathbb{R}^8$ defined by vertices <1,0,0,...,0>, <0,1,0,..,0>,...,<0,...,0,1>. A section is a 3-dimensional linear subspace of $\mathbb{R}^8$ that ...
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Which (semi)regular polyhedra are combinations of two others?
The convex combination of convex polytopes is a convex polytope.
An example in $\mathbb{R}^2$ is that a regular octagon
can be obtained as $\frac{1}{2} S + \frac{1}{2} S'$,
where $S$ is a square and $...
- 151k
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Nice proof of the triangle inequality for the metric of the hyperbolic plane
I am writing something for the journal of the university on the Lorentz space, and I want to prove that the following definition of the hyperbolic distance on the upper sheet of the hyperboloid $v \...
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Connecting a compact subset by a simple curve
Let $K$ be a compact subset of $\mathbb R^n$ with $n\ge 2$ (say if you like $n=2$, which is possibly sufficiently representative).
Q: Does there exist a closed simple curve $u:\mathbb S^1\to\mathbb R^...
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Triangle with largest perimeter in a convex region
What is the largest value of $r$ such that the following statement is always true?
"Let $C$ be a convex region with area $1$. There must exist a triangle contained in $C$ whose perimeter is at least ...
- 111
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Metrically homogeneous subsets of the plane
A metric space $M$ is metrically homogeneous if for every pair of points $x, y \in M$ there is an isometry $f$ of $M$ onto $M$ such that $f(x)=y$. What is known about metrically homogeneous spaces? ...
- 7,276
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Levy's isoperimetric inequality for sphere
Let me recall subj:
If $s>0$, $A$ and $B$ are two subsets of $\mathbb{S}^{n}$, $|A|=|B|$ ($|\cdot|$ stands for the Lebesgue measure on the sphere) and $B$ is a cup $B=\{ (x_1,x_2,\dots,x_n)\in \...
- 109k
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Midpoint geodesic polygon / Birkhoff curve shortening
I would like to know under what conditions the process
of creating a midpoint piecewise geodesic polygon converges
on a surface $S \subset \mathbb{R}^3$.
$S$ may be assumed smooth, closed, and ...
- 151k
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Algorithm for embedding a graph with metric constraints
Suppose I have a graph $G$ with vertex set $V$, edge set $E \subseteq {V \choose 2}$, a poistive integer $d$, and a weight function $w:E \to \mathbb{R}^{+}$. Is there a nice algorithmic way to decide ...
- 7,857
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Weighted Jaccard Similarity
I'd like to calculate the similarity between two sets using Jaccard but temper the results using the relative frequency of each item within a corpus.
Jaccard is defined as the magnitude of the ...
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Three half circles on the plane may not meet nicely
Let $H$ denote the union of the northern hemisphere of the unit circle $S^{1}$ with the interval $[-1,1]$ on the $x$-axis. That is, $H=\{(x,\sqrt{1-x^{2}}):-1\le x\le 1\}\cup\{(x,0):-1\le x\le 1\}$
...
- 2,136
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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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A generalization of the law of tangents
The law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides.
Let $a$, $b$, and $c$ be the lengths of the three ...
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Balls in Lawvere metric spaces
Let $V$ be the monoidal category $[0,\infty)$ (as a poset) with $+$ and $0$. Lawvere shows that $V$-enriched categories are a more natural generalisation of the notion of a metric space (note no ...
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Is the center of gravity in a CAT(0) space contained in the convex hull?
In reading Greg Kuperberg's partial answer to this question Convex hull in CAT(0) ,
I started wondering if the center of gravity is always contained in the closed convex hull.
More precisely, given $...
- 10.1k
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The number of relevant scales for a finite metric space
For an $n$-element metric space $X=\{x_1,\dots,x_n\}$ with metric
$d$ we introduce an array containing $\frac{n(n-1)}2$ numbers
$d(x_i,x_j)$, $i<j$. We assume that all distances are at least
$1$. ...
- 4,878
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Which surfaces admit unbounded-length simple geodesics?
Let $S$ be a surface embedded in $\mathbb{R}^3$.
A simple geodesic on $S$ is one that does not self-intersect.
Some surfaces have simple geodesics whose length exceeds any
given bound $L$. For example,...
- 151k
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Isometric imbedding of a sphere with positively curved metric
QUESTION. Given a Riemannian metric on the sphere $S^n$ with positive sectional survature. Can it be isometrically imbedded into $\mathbb{R}^{n+1}$ (of any class of regularity) as a boundary of a ...
- 21.8k
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Could a perfect squared square be split into two perfect squared squares?
This is a geometric puzzle though it might conceivably
also define a special class of Pythagorean triples.
A perfect squared square PSS is a square (as a plane figure)
partitioned into smaller ...
- 1,375
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The intersection of a circle and a rank 3 subgroup of the plane
Let $A$ be a rank 3 subgroup of the Euclidean plane, i.e. $A = \mathbb{Z} v_1 + \mathbb{Z} v_2 + \mathbb{Z} v_3$, where $v_1, v_2, v_3 \in \mathbb{R}^2$ are three $\mathbb{Q}$-linearly independent ...
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