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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On Gromov's proof of the systolic inequality $\operatorname{Sys}_1(M)\leq 6\operatorname{FillRad}(M)$
In the page 10 of the paper "Filling Riemannian manifolds" by Gromov (ProjetEuclid link), the author proves the following inequality (1.2) relating the systole and the filling radius of manifolds.
$$\...
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Estimation of the Gromov–Wasserstein distance of spheres
Let $(X,d_X,\mu_X)$ and $(Y,d_Y,\mu_Y)$ be two metric measure spaces. A probability measure $\mu$ over $X\times Y$ is called a coupling if $(\pi_1)_\sharp \mu=\mu_X$ and $(\pi_2)_\sharp \mu=\mu_Y$. We ...
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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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On bi-invariant metrics on groups
I'm looking for a quick, snap-your-fingers proof of the following result:
A continuous length metric on $\mathbb{R}^n$ that is invariant under translations comes from a norm.
To be clear about the ...
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Maximum number of Vertices of Hypercube covered by Ball of radius R
Let $R>0$ be given and let $H^n$ be the unit hypercube in $\mathbb{R}^n$. The problem I am facing is to find the maximum number of vertices of $H^n$ which can be covered by a closed $n$-dimensional ...
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Chameleon Bodies
(Note: This was orginally posted on New Year's Eve, 2011.)
Let a body $B$ be a compact set in $\mathbb{R}^3$ with a piecewise smooth boundary.
Some pieces/patches of the boundary are perfect mirrors; ...
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Are packing-homogeneous spaces homogeneous?
Given a metric space (M,d) define the packing function P(x,R,r) to be the maximum number of non-intersecting balls of radius r with centers in the ball B(x,R). Let’s call M packing-homogeneous if the ...
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Vietoris-Rips complex and coarse geometry
Let $K$ be an infinite countable subset of Euclidean space $E$ such any point of $E$ is within distance 1 of some point of $K$. In the language of John Roe's "coarse geometry", this implies that $K$ ...
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Open (resp., closed) balls homeomorphic to open (resp., closed) discs on the plane
Let $\Sigma$ be a compact (smooth) surface, with a geodesic metric $d$ (compatible with the topology of $\Sigma$).
Let $x \in \Sigma$, and suppose you have the following: for every $r<1$,
the open ...
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Stochastic Covering Number of a Convex Set
Consider a convex set, say $S = [0,1]^d$. Let $X_1, X_2,\ldots,X_n, \ldots$ be i.i.d. random variables that are uniformly distributed on $S$. Denote the Euclidean ball centered at $x \in \mathbb{R}^d$ ...
- 1,127
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Conic neighborhoods ⇔ polyhedral
I am looking for a reference to the following fact (I can prove it
my-self, but it should be known for a century).
Let $X$ be a reasonable metric space such that each point has a spherical
...
- 466
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Spanning curves by flat surfaces
Given a smooth closed connected curve $\gamma$ in $\mathbb R^3$, is there an immersed surface $S$ with boundary, such that its Gaussian curvature is equal to zero and $\partial S=\gamma$?
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Minimum reflection paths in a mirror polygon
Let $P$ be a simple, orthogonal polygon of $n$ edges, i.e., one whose edges meet at right angles,
and is non-self-intersecting;
also known as a rectilinear polygon.
Treat every edge of $P$ as a ...
- 151k
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Are aperiodic monotiles generalizable to higher dimensions?
This question is motivated by a recently released paper written by David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. It constructs the first topological disk that tiles the ...
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Do cut-length-minimizing equidissections exist?
Suppose $A,B$ are polygons of equal area. By the Wallace-Bolyai-Gerwien theorem, $A$ and $B$ are equidissectable: we can make finitely many straight-line cuts in $A$ and rearrange the resulting pieces ...
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Kissing the Monster, or $196,560$ vs. $196,883$
The $D = 24$ kissing number is $196,560$, and the dimension of the smallest non-trivial complex representation of the Monster group is $196,883$. These two numbers are nearly but not quite equal, and ...
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Metrization of projective manifolds
A modern take on Hilbert's fourth problem could be as follows:
Given a manifold $M$ with a flat projective structure (i.e., a $(PGL(n+1),\mathbb{RP}^n)$-structure), find all metrics for which the ...
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Topological dimension, Hausdorff dimension, and Lipschitz mappings
I can prove the following result. Here $\operatorname{dim} X$ stands for the topological dimension and $\mathcal{H}^n$ denotes the Hausdorff measure.
Theorem. Suppose that $f:\mathbb{R}^n\supset\...
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Does a rank 1 CAT(0) space with a proper cocompact group action contain a zero width axis?
A geodesic in a proper CAT(0) space is said to be rank 1 if it does not bound a flat half-plane and zero-width if it does not bound a flat strip of any width.
Let $X$ be a geodesically complete CAT(0) ...
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A lattice with Monster group symmetries
The book Mathematical Evolutions contains the following excerpt:
A last, famous, example is the following. It is known that in the space
of one hundred and ninety six thousand eight hundred and ...
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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
...
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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:=...
- 42k
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Boomerangs in Polya's orchard
Polya's orchard problem asks for what radius $r$ of trees
at each lattice point within a distance $R$
of the origin block all lines of sight to the exterior of the orchard.
The answer is known; $r$ ...
- 151k
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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
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diameter as a Morse function
Consider the space $X_1$ of closed subsets not containing a pair of antipodal points of the unit circle. Here we have a kind of degenerate Morse function, defined by the diameter of the pointset. ...
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Surfaces with many (but not solely) closed geodesics?
Let $S$ be a closed surface embedded in $\mathbb{R}^3$,
let's say of genus zero.
I seek examples of $S$ with the following property:
If one selects a random any point $p$ on $S$, and a random
...
- 151k
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Bi-spherical polyhedra
Bicentric polygons have been studied: a polygon all of whose vertices lie on its
circumcirle, and whose incircle is tangent to every edge:
I have not been able to find a comparable literature ...
- 151k
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Rectangology and squareology
I thought that rectangles were simple, and squares even simpler. Until my research has led me to several questions about rectangles and squares, which I can't solve.
I started by posting this question ...
- 3,549
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Is it overkill to invoke Kirszbraun theorem to prove the following fact ?
Given a small enough convex triangle $(abc)$ in a (smooth or Alexandrov) surface $(X,d)$ of curvature greater than $-1$, let $(\overline{abc})$ be its comparison triangle in $\mathbb{H}^2$. Then there ...
- 4,123
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Interpolating points with minimum curvature constraint
I have $n$ points $p_i$ strictly interior to a rectangle $R$,
and I would like to connect them with a curve $C$ whose curvature is as low as possible.
Let $\kappa_\max(C)$ be the sharpest (largest ...
- 151k
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is a group $G$, that admits finite $k(G, 1)$ and has no Baumslag-Solitar subgroups, necessarily hyperbolic?
This is the first question asked in Bestvina's article "Questions in Geometric Group Theory". Does anyone know if there has been any progress made on this problem? Is the question answered if $G$ is ...
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Dissecting trapezoids into triangles of equal area
[Lightly edited for copy and proper formatting of mathematics. -- Pete L. Clark]
The Background: Let $T$ be a trapezoid. Sherman Stein, using valuation theory, showed that if $T$ is dissectible into ...
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Feasibility of a list of prescribed distances in R^3
I am puzzled with the following problem:
Given $n$ real numbers it is to obtain a Yes/No answer to: "whether it is possible to arrange different points in the Euclidean $\mathbb{R}^3$ so that every ...
- 555
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631
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Estimating direction from a distribution on a circle
Let there be $n$ points on a unit circle. It is known they come from "normal" distribution around particular unknown direction (i.e. sum of 2 "normal" distributions on circle - one centered at point $...
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Self-Intersection of closed curves
Supoose I have a closed curve $\gamma$ in the plane such that for any isometry $g$ of $\mathbb{E}^2,$ such that $g(\gamma)\neq \gamma,$ $\gamma$ intersects $g(\gamma)$ in at most two points. It ...
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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 ...
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Classification of surfaces composed of circles
Define a circle as a geometric circle of positive, finite radius:
a set of points in $\mathbb{R}^3$ congruent to the
set $x^2 + y^2 = r^2$ in the $xy$-plane. [Edited as per BMann's comment.]
I am ...
- 151k
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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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Shortest irrational path
What is the shortest curve $\gamma$ in $\mathbb{R}^2$
from the origin $o=(0,0)$ to a rational point $p=(a,b)$
that (a) passes through no other rational point, and
(b) contains no point a ...
- 151k
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Can one do without a classifying space when showing vanishing of cohomology
Let $G$ be a discrete group and $A$ an abelian group, then $H^n (G,A)$ can be defined as $$ H^n (G,A) = H^n (B_G, A)$$
Where $B_G$ is the classifying space of $G$, i.e. $B_G = E_G / G$ where $E_G$ is ...
- 1,163
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Triple bubble conjecture: Natural candidate?
Is there a standard natural candidate surface for
the shape that encloses three given volumes
in $\mathbb{R}^3$ and has minimal surface area?
I know the planar triple bubble conjecture was proved
...
- 151k
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Approximation by locally Lipschitz functions
Could you tell me what is the name and/or reference for the following theorem:
Let $M$ be a metric space. Then any continuous function $f:M\to\mathbb R$ can be a be uniformly approximated by a ...
- 1,785
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Generalization of Sylvester-Gallai theorem
The Sylvester-Gallai theorem states that it is not possible to arrange a finite number
of points so that a line through every two of them passes through a
third unless they are all on a single ...
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420
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About a solid which satisfies $\sum_{i=1}^{n}x_i=0, |x_i|\le1\ (i=1,2,\cdots,n)$
For $n\ge 2\in\mathbb N$, let $S_n$ be the volume of a $(n-1)$ dimensional solid which satisfies
$$\sum_{i=1}^{n}x_i=0, |x_i|\le1\ (i=1,2,\cdots,n).$$
Then, here is my question.
Question : Can we ...
- 4,757
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Books about capacity theory
While I was studying the book Variation et Optimisation de formes by Antoine Henrot and Michel Pierre, I encountered a section about the capacity associated to the $H^1$ norm, which is defined for ...
- 2,222
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Recognising group actions on trees from the boundary
Let $G$ be a group acting on a locally finite tree $T$. Then the boundary $\partial T$ is a Cantor set on which $G$ acts by homeomorphisms (indeed by quasi-isometries under a suitable metric). ...
- 4,728
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Characterizing a tumbling convex polytope from the surface areas of its two-dimensional projections
My general question concerns what we can learn about an arbitrary, three-dimensional convex polytope (or convex hull of an arbitrary polytope) strictly from the surface areas of its two-dimensional ...
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586
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Unknown work of Nöbeling on topological/Hausdorff dimension
Let $\mathcal{H}^n$ denote the Hausdorff measure, $\dim_H X$ the Hausdorff dimension, and $\dim X$ the topological dimension of $X$.
A well known result of
Szpilrajn (He changed his name to ...
- 28k
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Generalized figures of constant width
Is it known which plane figures $Q$ can rotate touching three given circles $A$, $B$, and $C$?
This question was asked by Lazar Lyusternik in 1946, there is only one reference to this paper that ...
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