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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Using mirrors to make a non-convex polygon visible from a fixed interior point
Take a point $A$ inside a non-convex polygon $P$. Is it always possible to place a finite set of mirrors given by straight segments (not necessarily along the boundary of $P$, any position inside $P$ ...
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Constructing a metric over a lattice
Consider a lattice $({\cal L}, \wedge, \vee)$ with an antimonotonic function $f: {\cal L} \rightarrow {\mathbb R}$ defined on it (i.e $x \preceq y \implies f(x) \ge f(y)$).
$f$ is said to be ...
- 4,462
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Why is GL(n,C)/U(n) a CAT(0) space?
The title says it all. In one of his answers to the question "Convex hull in CAT(0)" (I don't have the points to post a link, if someone doesn't mind link-ifying this that would be cool), Greg ...
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Reference request: acceleration/curvature of curve in metric space
Let $(X,d)$ be a metric space. Given a continuous curve $\gamma_t : [0,1] \rightarrow X$, the metric speed is defined by $$ |\gamma_t^\prime | := \lim_{s\rightarrow t} \frac{d(\gamma_s, \gamma_t)}{|t-...
- 660
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Characterization of metrics such that the volume of balls doesn't depend on their centers?
Given a finite metric space $(X,d)$,
when does it hold that for all $y\in X$ and $r>0$, $\#B(y,r)$ does not depend on $y$?
Here $ B(y,r):=\{x\in X: d(x,y)\le r\} $ denotes a ball of radius $r$ ...
- 161
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Compact manifolds locally bi-Lipschitz to Euclidean space
I have a compact manifold $M$, and I am allowed to choose some Riemannian metric on it, exactly which I don't care. But I would love it if I could choose the metric $g$ such that every point has an ...
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Most dispersed set of points in a disk?
Put $1$ billion points in a disk of radius $1$. Consider the minimal area $A$ of a triangle formed by any $3$ points. Where do you put the points so that $A$ is maximal and how much is $A$?
Consider ...
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Hilbert space compression of the lamplighter group
What is the Hilbert space compression exponent of the standard lamplighter group $\mathbb{Z_{2}} \wr \mathbb{Z}$? For $\mathbb{Z} \wr \mathbb{Z}$ it is known to be $2/3$ by work of Austin, Naor and ...
- 2,449
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Cylinders dividing $\mathbb{R}^{3}$
Consider $n$ affine copies of a compact cylinder, say $S^{1}\times [-3,3]$ with top and botom, sitting inside $\mathbb{R}^{3}$.
For each $n$ we may ask ourselves how to arrange the $n$ cylinders so ...
- 2,136
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Stability of midpoints in CAT(0) spaces
Given a CAT(0) space $X$ and a compact, convex subset $A$ of $X$. One can define its midpoint $m(A)$ as the point, at which the following function attains its minimum.
$f:A\rightarrow \mathbb{R}\qquad ...
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Are finitely generated subgroups of $\text{GL}_n(\mathbb{Q})$ virtually special?
This might be a silly question--but are there any examples of finitely generated subgroups of $\text{GL}_n(\mathbb{Q})$ that are known to not be virtually special?
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Is there an L-system for aperiodic tilings of the plane with the "hat" monotile?
Most aperiodic tilings of the plane, except possibly for spiral tilings like the Voderberg tiling, exhibit a fractal pattern of self-similarity. This is no exception for the recently discovered "...
- 13.4k
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On planar sections of 3D convex bodies
Consider the space of planar sections of any given convex 3D body.
Basic Question: What is the lower bound for the ratio
$$\frac{\text{area of section of greatest perimeter}}
{\text{area of section of ...
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Necessary and sufficient condition for tangential polygon to be cyclic
Can you prove or disprove the following claim?
Claim. Let $A_1,A_2, \ldots ,A_n$ be the vertices of an $n$-sided tangential polygon and let $B_1,B_2, \ldots ,B_n$ be the contact points of the ...
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Cutting the unit square into pieces with rational length sides
The following questions seem related to the still open question whether there is a point(s) whose distances from the 4 corners of a unit square are all rational.
To cut a unit square into n (a finite ...
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Does a minimum area disk that is bounded by a cycle $C$ continuously deform in $R^3$ as $C$ moves in $R^3$?
Let $C_1=(v_1,v_2,\ldots,v_{i-1},v_i)$ and $C_2=(v_1,v_2,\ldots,v_{i-1},v'_i)$ be two cycles that are drawn in $R^3$ in the shape of an unknot (not knotted) with straight line segments as their edges (...
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A (possibly boring) Voronoi Game
The board for this game is a compact convex region $\cal C$ of $\mathbb{R}^2$.
Below I illustrate with $\cal C$ an equilateral triangle.
Two players, $A$ and $B$, alternate turns.
At each turn they ...
- 151k
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The Universality Theorem by Mnev for uniform oriented matroids of rank 4 and higher
According to the Universality Theorem by Mnev (see below theorem 8.6.6 from [1]), for any open semialgebraic variety V there is a uniform oriented matroid of rank 3 whose realization space is stably ...
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Reverse plane geometry, anyone?
I refer to Greenberg's wonderful 2010 MAA article "Old and new results in the foundations of elementary plane Euclidean and non-Euclidean geometries". There, and in his book, Greenberg defines a ...
user2529
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Intersecting Sets of Pythagorean Triples with Common Hypotenuses
For any $r\in\mathbb{N}$, let $A_r$ denote the set of all natural numbers that are potentially a side of a Pythagorean triple with hypotenuse $r$.
Given any $N\in\mathbb{N}$, does there exist $r,s$ ...
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Number of neigbour Voronoi cells for a random set of points on S^k or cube [-1, 1]^k?
Consider $S^k \subset R^{k+1} $. Sample $N$ points by say uniform distribution. (Example k=120, N=2^24, i.e. N>>k ).
Consider Voronoi cell around each point.
How many neighbours would a cell have ...
- 24.9k
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710
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Pursuit-Evasion on a Manifold
I know pursuit-evasion has been studied in many contexts, including
on a manifold (e.g., Melikyan,
"Geometry of Pursuit-Evasion Games on Two-Dimensional Manifolds"),
but I have not seen this version:
...
- 151k
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Finding the convex combination of vertices which yields an inner point of a polytope
Given a convex polytope $P\in \mathbb{R}^n$, and a point $x\in P$, Caratheodory's theorem gives us that there exists a set of at most $n+1$ vertices of $P$, such that $x$ is a convex combination of ...
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Bending an elastic, inextensible sheet of paper into a teardrop shape
I take an rectangular sheet of paper, of height $H$ and Young's Modulus $E$, and in the absence of gravity, I bend it into a "teardrop" shape so that the edges along the top and bottom touch only ...
- 63
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Average squared distance vs diameter in vertex-transitive graphs
Let $X=(V,E)$ be a finite, connected graph on $n$ vertices, endowed with its graph metric $d$. The average squared distance of $X$ is $avg(d^2)=\frac{1}{n(n-1)}\sum_{x,y\in V,x\neq y} d(x,y)^2$; it ...
- 11.1k
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450
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Dissecting a tetrahedron into orthoschemes
Is there a way to dissect any tetrahedron into a finite number of orthoschemes?
I know that for a tetrahedron which only has acute angles, one can take the center of the inscribed circle and project ...
- 601
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1k
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Quantitative questions about the size of a finite epsilon net
Let $X$ be a metric space, and let $U \subset X$ be any set. A finite set $N = N(\epsilon) \subset U$ is called a finite $\epsilon$-net of $U$ if every point of $U$ is at most a distance of $\epsilon$...
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Locally compact + two-point homogeneous => Riemannian
A metric space $M$ is called two-point homogeneous if for any pair of points $(p,q)$ in $M$ any distance preserving map $f\colon\{p,q\}\to M$ can be extended to an isometry $\bar f\colon M\to M$.
The ...
- 45k
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Realizing spherical complexes as convex polytope
A spherical polytope is the intersection of some closed hemispheres which is non-empty and does not contain a pair of antipodal points. A spherical complex is a tiling of the whole (d−1)-dimensional ...
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Isometric imbedding of a 2-disk into Euclidean 3-space
Let us call a cap the intersection of the boundary of 3-dimensional convex compact set $K$ in $\mathbb{R}^3$ with a half-space bounded by a plane $H$ such that the orthogonal projection to $H$ of this ...
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Average distance of the mean of $n$ random complex numbers in a unit disc
Let $z_1,z_2,\dots,z_n$ be $n$ complex numbers distributed uniformly and randomly over the unit disc $x^2+y^2 \leq 1$. Let $z$ be the complex number defined by the mean of the of these numbers,that ...
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Rectangles in rectangles and $(b^2-a^2)^2\le (ax-by)^2+(bx-ay)^2$
When does an $a\times b$ rectangle fit inside an $x\times y$ rectangle? I have an algebraic condition which I can diagram geometrically, and I'd like a good geometric argument.
Assume $0<a<b$, $...
user44143
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Is a symmetric, parallel (0,2)-tensor a metric?
I'm interested in affinely connected spaces, on which a metric is not necessarily defined, i.e. $(\mathcal{M},\Gamma)$. Since (as a physicist) my goal is to consider a generalized model of gravity, I ...
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Contractibility of balls in Alexandrov spaces
Let $X$ be a compact finite dimensional Alexandrov space with curvature bounded below.
Does there exist $\varepsilon_0>0$ (depending on $X$) such that for any $\varepsilon \in (0,\varepsilon_0)$...
- 21.8k
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Norms on $\mathbb{R}^d$ whose linear isometries are the hypercube group
It is a known fact that for any $2\neq p\in[1,\infty]$, the linear isometries for the corresponding norm $\|\cdot\|_p$ on $\mathbb{R}^d$ is the set of all square-matrices with entries in $\{-1,1,0\}$, ...
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Motivation for Hirzebruch-Jung Modified Euclidean Algorithm
Let $a,b \in \mathbb{N} \ \ s.t. \ \ a > b$ have $\gcd(a,b) =1$. We can define the Hirzebruch-Jung modified euclidean algorithm as follows:
Let $e_i \in \mathbb{N} >2$, and $ r_k \in \mathbb{N}$...
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Existence of a measurable map between metric spaces
Let $X$ and $Y$ be separable complete metric spaces (if necessary, they may be assumed to be compact). Let $R\subset X\times Y$ be a closed subset such that the projection of $R$ to $X$ is onto.
Is ...
- 21.8k
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local quasi geodesics in hyperbolic spaces
I asked this question on math stackexchange (see here) but didn't get any answer so I thought I post it here too.
We have the following two well-known Theorems:
T1) For all $\delta > 0, \lambda ...
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Which criteria guarantee an orthogonal circuit in $\mathbb R^3$ to be rigid?
For $n\ge4$, define an orthogonal circuit or O-circuit as a closed circuit of $n$ unit segments in $\mathbb R^3$ such that any two neighboring segments form a right angle. (Physically this could be ...
- 13.4k
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639
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Local geodesics in uniquely geodesic spaces
A while ago I asked this
question in Math Stackexchange. Since I didn't receive an answer so far, I thought I'd ask it here.
Suppose $Y$ is a proper length space, where every pair of points $x,y\in Y$...
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Are shortest halving curves simple closed geodesics?
Let $S$ be a smooth convex surface in $\mathbb{R}^3$
(although my question may as well be asked for the surface of a polyhedron).
Say that $\gamma$ is a shortest halving curve if
(a) it partitions the ...
- 151k
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Taylor expansion of the determinant of a Riemannian metric
Let $(M,g)$ be a compact Riemannian manifold without boundary. Fix a point $x\in M$ and $N\ge 2$ large. Then there exists a metric $\tilde g$, conformal to $g$ such that $$ \det \tilde g=1+O(r^N)$$ ...
- 327
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Rationality of translation lengths in hyperbolic groups
Recall that the translation length $\tau(g)$ of an element $g \in G$ is the limit $d(1, g^n)/n$, where $d$ is the word metric on $G$ with resepct to some generating set.
It is a theorem of Gromov ...
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Passing C through a slot
Question: Given a closed curve C, what will be the (bounds on) dimension of the interval it will pass through?
i.e. which are the necessary and sufficient conditions for a planar compact set C to pass ...
- 851
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Does an origin-centered ellipse in the plane intersect each $L^p$-circle at most 8 times?
The question is in the title: Let $E$ be an origin-centered ellipse in ${\mathbb R}^2$ and let $S$ be an "$L^p$-circle": $S = \{(x,y) : |x|^p + |y|^p = \text{const}\}$, where $1 \leq p \leq \infty$. ...
- 6,694
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Lattice-cube minimal blocking sets
Let $C_d(n)$ be the lattice cube consisting of the $n^d$ points with
each of its $d$ coorindates in $\lbrace 1,2,\ldots,n \rbrace$.
Define a blocking set for a lattice cube to be a set of points
in ...
- 151k
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Examples of CAT(0)-groups
My question is the following:
Let M be a simply connected Riemannian manifold whose sectional curvatures
are all nonpositive and let G be a group. Suppose that G acts in M properly discontinuous and
...
- 235
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2
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215
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Partition of polygons into 'strongly acute' and 'strongly obtuse' triangles
Definition: Let us refer to obtuse triangles with the largest angle strictly above a given cutoff value as 'strongly obtuse' - the definition is parametrized by the cutoff value. Likewise, strongly ...
- 5,979
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The algebraic structure of a line in a (Tarski) plane
By a Tarski plane (resp. plane) I understand a mathematical structure $(X,B,\equiv)$ consisting of a set $X$, a ternary betweenness relation $B\subseteq X^3$ and the 4-ary congruence relation ${\equiv}...
- 42k
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Can every smooth space curve be realized as an origami curved crease?
Many years ago, Ron Resch told me that he proved that every smooth simple space curve $C$ could be realized as a curved crease $\gamma$ in the interior of a piece of paper.
He never published this (as ...
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