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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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 ...
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Why is proving $C^{\infty}$ regularity of sub Riemannian geodesics so hard?
In Montgomery's A Tour of Subriemannian Geometries, Their Geodesics and Applications, problem 10.1 in Chapter 10 asks "Is every minimizing geodesic smooth ?".
Can someone explain what are the major ...
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Ellipsoids and lattices: an enclosure problem.
$E \subset {\mathbb R}^2$ is an ellipse of area $1$ centered at the origin that contains no other point with integer coordinates. Is there a matrix $A \in SL(2,{\mathbb Z})$ such that the ellipse $A(E)...
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Metric spheres in CAT(0) manifolds
Let $X$ be a topological manifold of dimension $n$, equipped with a compatible CAT(0) metric.
Are sufficiently small metric spheres in $X$ homeomorphic to metric spheres in Euclidean space $\mathbb{E}^...
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Largest convex hull of a unit length path
What is the largest area possible for the convex hull of a path of unit length lying on a plane? For what paths is that largest area attained?
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Complexity of matching red and blue points in the plane.
I'm just asking because I'm curious.
I was seeking references on the following problem, that a friend exposed to me last holidays :
Problem
Given $n$ red points and $n$ blue points in the plane in ...
- 4,123
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Concentration of measure for arbitrary convex bodies?
There are various "concentration-of-measure" theorems,
the best known that due to Lévy,
which is this (informally): the volume of a sphere $S^d$ in $d$ dimensions is largely
concentrated around an $\...
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What are trig classes like within a universe that's "noticeably" hyperbolic?
[I want to think that this question has an answer, but it may be more a "community wiki" discussion. Feel free to re-tag.]
What are trig classes like within a universe that's "noticeably"[*] ...
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Non-Kahler manifolds where the different Laplacians are compatible
On a Kahler manifold, the different Laplacians are compatible: $\Delta_d=2\Delta_{\bar{\partial}}=2\Delta_{\partial}$.
Are there non-Kahler Hermitian manifolds where the above identity holds?
user2529
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Mutually tangent ellipsoids in 3 space
I recently heard a claim that for any n, it is possible to arrange n ellipsoids in 3 space such that each pair of ellipsoids is kissing. Is this true, and if so, how?
Edit: By kissing, I mean that I ...
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Determine if you can build a polygon from segments [closed]
Is there a way to determine whether it is possible to build a polygon from given n segments?
Maybe triangle inequality generalized?
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Gromov hyperbolic groups which are solvable are elementary
I have read on wikipedia that a Gromov hyperbolic group which is solvable is elementary (i.e. virtually cyclic). Where can I find a proof of this fact?
There is a proof of a similar fact in Bridson-...
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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 \...
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Is there a straightedge and compass construction of incommensurables in the hyperbolic plane?
In other words, given a segment in the hyperbolic plane is there a straightedge and compass construction of a segment incommensurable with it? In the Euclidean plane one can take the diagonal of the ...
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Isoperimetry and Poincaré Inequality
What are the known relations between isoperimetric and Poincaré inequalities on manifolds?
For example, for manifolds with a lower bound on Ricci curvature, the Cheeger-Buser inequality relates the ...
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Distance function to a submanifold
Let $M$ be a compact Riemannian manifold and $\Sigma\subset M$ a closed submanifold. Given $x\in M$ we define the distance function to $\Sigma$ by $$d_\Sigma(x):=\inf\{d(x,y):y\in \Sigma\},$$ where $d$...
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Is it always possible to "encircle" exactly $n$ points in an infinite subset of $\mathbb{R}^d$ without limit points?
Let $d$ be a positive integer, and let $\mathbb{R}^d$ be endowed with the
Euclidean metric. Given an infinite set $S \subset \mathbb{R}^d$ without limit points
and a positive integer $n$, is there ...
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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}{\...
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Minimize Perimeter(S)/Area(S) for S inside the unit square.
This is a very silly question.
For all regions S contained inside the unit square, what is the infimum of the quantity Perimeter(S)/Area(S)? This ratio being considered is not scale invariant, so it ...
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Tracking a reference: "Karl Scherer, A Puzzling Journey to the Reptiles and Related Animals"
I linked a paper by James Schmerl in a recent question which cites Karl Scherer, A Puzzling Journey to the Reptiles and Related Animals, Privately Published, 1987.
I have had difficulty finding any ...
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Equational theory of the orthocenter
Previously asked at MSE:
Briefly speaking, I'm looking for a description of the equational theory of the orthocenter function, $\mathsf{orth}$. By $\mathsf{orth}$ I mean the (partial) function sending ...
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Is a polytope that has in-spheres for faces of all dimensions already regular?
Let $P\subset\Bbb R^d$ be a convex polytope (convex hull of finitely many points).
A $k$-in-sphere of $P$ is a sphere centered at the origin to which each $k$-face of $P$ is tangent. So a 0-in-sphere ...
- 13.6k
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Other norms for lattice reduction techniques (LLL, PSLQ)?
LLL and other lattice reduction techniques (such as PSLQ) try to find a short basis vector relative to the 2-norm, i.e. for a given basis that has $ \varepsilon $ as its shortest vector, $ \varepsilon ...
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Bi-partitioning $2n$ points on the plane with a straight line
Let $S$ be a set of $2n$ points in $\mathbb{R}^2$. Which is the maximum number of different bi-partitions of $S$ generated by a straight line?
More precisely, which is the maximum number of partitions ...
- 4,459
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The devil's playground
On the $\mathbb{R}^2$ plane, the devil has trapped the angel in an equilateral triangle of firewalls.
The devil
starts at the apex of the triangle.
can move at speed $1$ to leave a trajectory of ...
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160
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Hyperplane arrangements whose regions all have the same shape
Suppose I have a (finite, real, central, essential) hyperplane arrangement $\mathcal{H}$ such that all regions "have the same shape": for any two regions $R,R'$, there is an orthogonal ...
- 2,753
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What does convergence in distribution "in the Gromov–Hausdorff" sense mean?
I am trying to understand this survey article by Le Gall on Brownian geometry, especially the statement of Theorem 1.
The basic statement of the theorem is
$$(m_n,d_n) \to (m_{\infty}, d_{\infty})$$
"...
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Function as sum of distances over a connected, compact metric space
If $X$ is a connected, compact metric space with distance function $d : X^2 \rightarrow \mathbb{R}^+$, is it true that there exists a positive real number $a$, dependent on $X$ and $d$, such that for ...
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Making compact subsets "parallel"
Let $X$ be a compact metric space. Say that two compact subsets $E,F\subset X$ are parallel if
$$ dist(x,F) = dist(y,E)$$
for all $x\in E$ and $y\in F$. Here $ dist(y,E) = \inf\{d(y,z):z\in E\}.$
The ...
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Subtlety in the definition of the Kobayashi metric
When defining the Kobayashi metric on a connected complex analytic space $X$, one makes the following auxiliary definition:
A holomorphic chain from $x\in X$ to $y\in X$ is a finite sequence of ...
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Isometries of convex hypersurfaces
The well known Pogorelov’s rigidity theorem says that if two convex closed surfaces in Euclidean 3-space are isometric to each other being equipped with their intrinsic metrics, then they are ...
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Rational points on circular spirals
Is it the case that every unit-radius circular spiral,
$$x = \cos(t)$$
$$y = \sin(t)$$
$$z = c \cdot t$$
for $c \in \mathbb{R}^+$
is dense in rational-coordinate points
(i.e., all three coordinates ...
- 151k
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Reference request: Ehrhart's conjecture on the geometry of numbers
Conjecture (Ehrhart). If a convex body $K \subset {\mathbb R}^n$ has its barycenter at the origin and contains no other point with integer coordinates, the volume of $K$ is less than or equal to $(n +...
- 13.5k
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Quasi-isometry classes of elementary amenable groups
Is there any elementary argument showing that there exist uncountably many distinct quasi-isometry classes of elementary amenable groups? How about solvable groups?
For amenable groups it follows ...
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Proving the inequality involving Hausdorff distance and Wasserstein infinity distance
Prove the inequality
$$d_{H}(\mathrm{spt}(\mu),\mathrm{spt}(\nu))\leq W_{\infty}(\mu,\nu)$$
where $d_H$ denotes the Hausdorff distance between the supports of the measures $\mu$ and $\nu$, and $W_\...
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When is "metric dimension" well defined?
A subset $B$ of a metric space $(M,d)$ is called a metric generating set if and only if $$[\forall b \in B, d(x,b)=d(y,b)] \implies x = y \,. $$ A metric generating set $B$ is called a metric basis ...
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Diameter of random segment intersection graph?
I have an even number of points $n$ randomly distributed (uniformly) in a disk.
Then the points are randomly connected to form $n/2$ segments, a perfect
matching.
Finally, I form the intersection ...
- 151k
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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 ...
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Weitzenböck Identity for $\Delta_{\bar{\partial}_E}$
This question is related to this MO question and this MSE question.
Let $E$ be a hermitian holomorphic vector bundle over a hermitian manifold $X$. The bundle $\bigwedge^{\bullet,\bullet}X\otimes E$ ...
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What makes a distance?
In the answers to my previous question, I learned that there are different concepts of distance, that is of distance-like functions with the usual metric being only the most popular and important one.
...
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Maximal number of connected components of complement to an affine plane real algebraic curve
Let $X$ be a (singular, reducible) affine plane real algebraic curve of degree $d$.
How we can estimate maximal number of connected components of it's complement in $R^2$ in terms of degree?
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Is there a common name for the complement of a metric space in its completion?
Is there a common name for the complement $\widehat{X} \setminus X$ of a metric space $X$ in its metric completion $\widehat{X}$? Since $X$ is not necessarily open in $\widehat{X}$, the term boundary ...
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Probability that randomly chosen balls have a nonempty common intersection
Fix some $0 < r < 1$. A collection of points $x_1, \dots, x_n$ are chosen independently and uniformly at random from the closed unit ball in $\mathbb R^d$.
What is the probability that the ...
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Isoperimetric dimension for any (metric) measure space?
$\newcommand{\v}{\operatorname{vol}}$The isoperimetric dimension is the maximum $d$ s.t.
$$\v(D)\leq C\cdot \v(\partial D)^{d/d-1}$$
for all open with smooth boundary $D\subset M$, differentiable ...
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Differentiability of distance to a closed convex set [closed]
Let $( \mathbb{R}^d, \| \mathbf{x}\|_2 )$ be a Euclidean Space. For any nonempty closed convex set $A\subseteq \mathbb{R}^d$, we define
\begin{align}
d(\mathbf{x}, A) = \inf \{ \| \mathbf{x} - \mathbf{...
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Dissecting Ramanujan´s Cuboid: 1729 = 19 x 13 x 7
Consider the cuboid of dimensions 19 x 13 x 7 whose volume is 1729, the Hardy-Ramanujan number. What is the least number of smaller cuboids into which it can be dissected so that the resulting pieces ...
- 3,707
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Oloid and sphericon: rolling develops entire surface
Wikipedia says that,
"The oloid is one of the only known objects, along with some members of the sphericon family, that while rolling, develops its entire surface."
Below are illustrations of ...
- 151k
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What is the shape of mathematical universe?
Shape? At the usual mathematical literature when we can discuss about the shape of a "space" that we have a kind of "topography" on it. For example a topology, metric, geometry, etc.
Note that for ...
user36136
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Transitive geodesics on closed surfaces of genus greater than one
A well-known result of Hedlund and Morse states that if a Riemannian metric on a closed surface of genus $g > 1$ has no conjugate points, then it carries transitive geodesics (i.e., geodesics whose ...
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Generalizing a square wheel to a body rolling on a surface
A square wheel rolling on a catenary road maintains the wheel center at a fixed
height, a well-known construction previously discussed on MO
(e.g.,
"Generalizing square wheels rolling on inverted ...
- 151k