Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
692 questions
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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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2
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Wasserstein distance in R^d from one dimensional marginals
This question occurred to me while I was reading Klartag's papers on central limit theorems for convex bodies.
Given probability measures $\mu$, $\nu$ on (the Borel $\sigma$-field of) $R^d$ with ...
11
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2
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685
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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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Interpret Fourier transform as limit of Fourier series
Let $V=\mathbb{R}^n$, $\Lambda_r=2\pi r \mathbb{Z}^n \subset V (r>0)$ a lattice; $V^*\cong\mathbb{R}^n$ the dual vector space of $V$, and $\Lambda_r^*=\frac{1}{2\pi r} \mathbb{Z}^n =\text{Hom}(\...
- 1,906
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1
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Polyominoes with double contact
Here is a problem which arose from an earlier question. I'll change the terminology but not the question: A polyomino is a region with a connected interior made by joining one or more unit squares ...
- 30.1k
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2
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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
11
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702
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Schoenberg's rational polygon problem
"A polygon is said to be rational if all its sides and diagonals are rational, and I. J. Schoenberg has posed the difficult question, ‘Can any given polygon be approximated as closely as we like by a ...
- 151k
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Sum of squared nearest-neighbor distances between points in a square
Let $\square_2=\{(x,y): 0\leq x, y\leq1\}$ be the unit square in $\mathbb{R}^2$. Take $n>1$ points $P_1, \dots, P_n\in\square_2$.
Denote the distances $d_j=\min\{\Vert P_k-P_j\Vert: k\neq j\}$, ...
- 43.2k
11
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1
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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
11
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2
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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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2
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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,...
- 65.4k
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In a locally CAT(k) space, does uniqueness of geodesics imply the lack of conjugate points?
A complete, simply connected Riemannian manifold has no conjugate points if and only if every geodesic is length-minimizing. I just realized that I don't know whether the same is true for a locally ...
- 32.4k
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2
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A characterization of metric spaces admitting a bi-Lipschitz embedding into a Hilbert space?
Theorem (??) derived in this MO-post from Schoenberg's theorem yeilds a "bipartite" characterization of metric spaces that admit an isometric embedding into a Hilbert space. This Theorem (??)...
- 41.9k
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2
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388
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Tangled random triangles: One giant component?
Suppose you have $n$ triangles whose corners are random points on a sphere $S$
in $\mathbb{R}^3$.
Viewing the triangles as built from rigid bars as edges,
two triangles are linked if they cannot be ...
- 151k
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Does a compact contractible metric space have a point that is fixed by all isometries?
Let $(X,d)$ be a compact and contractible metric space. Let $\operatorname{Isom}(X)=\{\phi\colon X\to X\}$ be its group of isometries.
Question: Is there a point $x\in X$ fixed by all $\phi\in\...
- 13.6k
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3
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Given the skeleton of an inscribed polytope. If I move the vertices so that no edge increases in length, can the circumradius still get larger?
Let $P\subset \Bbb R^n$ be an inscribed convex polytope, that is, all its vertices are on a common sphere of radius $r$.
Let $G$ be the edge-graph of $P$. For convenience, assume $V(G)=\{1,\dotsc,s\}$....
- 13.6k
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2
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280
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Monochromatic point sets in two-colored plane
Which are the configrations $P\subset \mathbb{R}^2$ of points, such that the following property holds:
Property M (for Monochromatic): Every two-coloring of $\mathbb{R}^2$ contains a monochromatic ...
- 10.7k
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4
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The distribution of the shortest path through $n$ points
In the big picture, I'd like to know: if I sample $n$ points uniformly at random in the unit square, what is the probability that the shortest path that visits each one of them is very small?
More ...
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Polyhedron not circumscribed about a sphere
Let $P$ be a polyhedron whose faces are colored black and white so that there are more black faces and no two black faces are adjacent. Show that $P$ is not circumscribed about a sphere.
My teacher ...
- 1,090
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3
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Mean maximum distance for N random points on a unit square
Following up on Mean minimum distance for N random points on a one-dimensional line and Mean minimum distance for N random points on a unit square (plane), I have the following questions.
Given N ...
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Intersections and curvature in the plane
Let $D$ be a nonempty compact convex plane region whose boundary is a smooth curve whose radius of curvature is at most 1 everywhere. Can the boundary of $D$ intersect a circle of radius 1 in more ...
- 19.7k
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343
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Bang's open question strengthening Tarski's planks problem
Tarski's Planks problem,
solved by Thøger Bang in 1951, says (in a simplified $\mathbb{R}^2$ version) that it requires
"planks" (parallel strips) of total width $\ge d$ in order to completely cover
a ...
- 151k
10
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1
answer
494
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Ping-pong progress through a quincunx
A quincunx or
Galton board consists of
staggered pegs from which ping-pong balls bounce and eventually display
a binomial / normal distribution in catch-bins. I am wondering if the
downward progress ...
- 151k
10
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2
answers
764
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Generalization of Stewart's theorem?
I'm curious about the generalization of Stewart's theorem to more dimensions. MathWorld mentions that there is a generalization done by Bottema, but I could not find much information on it. All I ...
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3
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What spaces have well known horofunctions?
Following Gromov, take a metric space $(X,d)$ and consider $C(X)/\mathbb{R}$ the set of continuous functions to $\mathbb{R}$ with the topology of uniform convergence on compact sets after taking the ...
- 4,304
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415
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Lipschitz homotopy groups
There is an extensive literature on Lipschitz homotopies of Lipschitz maps. But I haven't seen anything about Lipschitz homotopy groups. We have introduced this notion in an article that you can find
...
- 28k
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441
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A new $\ell_p$-metric on the hyperspace of finite sets?
Let $(X,d)$ be a metric space and $Fin(X)$ be the family of all non-empty finite subsets of $X$. For every $n\in\mathbb N$ the elements of the power $X^n$ are thought as functions $f:n\to X$ where $n:=...
- 41.9k
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683
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Circles avoiding rational points of height $\le h$
Q. Which origin-centered circles $C(r)$ (or spheres in dimension $d$)
of radius $r < 1$ avoid all rational points
of height $\le h$?
A rational point is a point all of whose coordinates are ...
- 151k
10
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1
answer
893
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Doubling space without Besicovitch covering theorem?
A metric space is doubling if any ball of radius $2R$ can be covered by $N$ balls of radius $R$ and $N$ is fixed once forever.
Is there an example of complete length-metric space which is doubling, ...
- 293
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2
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797
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Fitting a mesh to a density function
Suppose I have a probability density function defined on a region in the plane (in my case, the pdf is of the form $f(x) = \alpha\|x\|^{-\beta}$, and the region is the unit disk). For large $N$, is ...
- 1,125
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1k
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Alexandrov angles in Riemannian manifolds
Dear all, I am teaching a course in Riemannian geometry, and I would like to prove some comparison theorems in the next lessons, building on the well-known theory of Jacobi fields, and of Rauch ...
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1
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545
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Is every metric continuum almost path-connected?
The question was motivated by this question of Anton Petrunin.
By a metric continuum we understand a connected compact metric space.
Let $p$ be a positive real number. A metric continuum $X$ is ...
- 41.9k
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2
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845
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Largest regular $k$-simplex inscribed in a $d$-cube, $k < d$
The largest (by edge length) regular simplex inscribed in a unit cube
is well known in $\mathbb{R}^2$ and $\mathbb{R}^3$:
Image sources:
left: NMSU,
right: Mathworld.
A recent Amer ...
- 151k
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1
answer
935
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Smoothability of compact Alexandrov surfaces with curvature bounded from below
Let $(X,d)$ be compact metric space of curvature greater than $-1$ (in the sense of comparison triangles), assume that its Hausdorff dimension is $2$. Then a result of Perelman says that $X$ is a 2-...
- 4,123
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If all balls around fixed basepoints are isometric, are the spaces as well (length spaces)?
Let $X$ and $Y$ be two complete proper length spaces, $x \in X$ and $y \in Y$. Assume for every $r>0$ the closed balls $\overline{B_r(x)}$ and $\overline{B_r(y)}$ are isometric.
Does there exist ...
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2
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846
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Minimum separation among $m$ random points on an $n$-dimensional unit sphere
Consider $m$ points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be
$$
\rho = \min_{i,j\in{...
- 1,127
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1
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557
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Length inequalities in trees and CAT(0) spaces
I have a family of possibly related questions. Let me start with an elementary one:
Question 1. Fix an integer $n$. For which collections of real numbers $a_{ij}$, $i, j = 1, \dots, n$, is it true ...
- 10.1k
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1k
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What rectangles can a set of rectangles tile?
(I asked this question first on math.stackexchange, but did not get any responses so I thought I would try here.)
If we have a set of $p_i \times q_i$ rectangles ($p_i, q_i \in \mathbf{N}$), which $m \...
- 431
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2
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Rational points on circular spirals
Is it the case that every unit-radius circular spiral,
$$x = \cos(t)$$
$$y = \sin(t)$$
$$z = c \cdot t$$
for $c \in \mathbb{R}^+$
is dense in rational-coordinate points
(i.e., all three coordinates ...
- 151k
9
votes
1
answer
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Uniform sampling from general simplex with a twist
This is part of a question I had asked elsewhere, and then some of the links redirected me to CS stack exchange.
Given $0\leq a_1\leq\dots\leq a_D\leq1$ (all strictly positive), I want to draw points ...
- 221
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1
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Billiard dynamics with angle of reflection a fraction of angle of incidence
Suppose that a billiard ball bouncing in a unit square (or a lightray reflecting
in a mirrored square) has the property that the angle of reflection is a fraction
of the angle of incidence, rather ...
- 151k
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5
answers
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Get a point inside a polygon
I have a 2D polygon of arbitrary geometry. I need to find any point that is inside of that polygon. Taking the center won't work, because the polygon might not be convex. Is there a way to quickly ...
- 201
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Is compass and straight edge geometry complete?
Euclid's first three postulates are the basis of compass and straight edge constructions which are as complex as arithmetic.
The constructions themselves may be expressed as a formula with each of the ...
9
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3
answers
1k
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Does there exist a notion of discrete riemannian metric on graph?
I would like to know if there is any notion of a discrete Riemannian metric on graphs. C. Mercat has worked on discrete Riemann Surfaces, but that's not exactly what I am working on.
To be more ...
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Thinnest covering of the plane by regular pentagons
Q. Is it known what is the thinnest covering of the infinite plane by regular pentagons?
By covering I mean every point of the plane is covered.
By thinnest I mean the proportion of the plane covered ...
- 151k
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votes
1
answer
457
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Best non-lattice sphere packings
Consider a dense sphere packing in $\mathbb{R}^n$, i.e. an arrangement of mutually disjoint solid open spheres, all of the same radius.
In dimensions $2, 3, 8,$ and $24$, it is known that lattice ...
- 1,046
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3
answers
623
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Line-preserving bijection of ${\mathbb{R}}^n$ onto itself
If $f:{\mathbb{R}}^n\to{\mathbb{R}}^n$ $(n\ge2)$ is a bijection such that the image of every line is a line (continuity of $f$ not assumed), must $f$ be an affinity?
Assuming continuity would ...
- 7,276
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240
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Cyclic polygons generalized to higher dimensions
Many theorems hold for cyclic polygons—convex polygons inscribed
in a circle. Perhaps the most basic is this,
from the reference cited below:
Theorem. There exists a cyclic polygon of $n \ge ...
- 151k
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2
answers
390
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Is it possible to continuously select a probability distribution over fixed points in Brouwer's fixed point theorem?
According to Brouwer's fixed point theorem, for compact convex $K\subset\mathbb{R}^n$, every continuous map $K\rightarrow K$ has a fixed point.
However, these fixed points cannot be chosen ...
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559
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What is the shape of the $n$-gon which gives the maximum of a function?
What is the shape of the $n$-gon $P_1P_2\cdots P_n$ which gives the maximum of $A_n$? The quantity $A_n$ is defined by
$$ A_n = \frac{{\sum_{i\lt{j}\le{n}}{\lvert P_i P_j\rvert}^2}-{\sum_{i=1}^{n}{\...
- 4,757