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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Injectivity radius on complete manifolds with positive and bounded curvature
I have two question:
1) Are there any examples of complete manifold with strictly positive and bounded section curvature which has zero injectivity radius?
2) Is there a sequence of non-compact ...
- 101
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513
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An isoperimetric-type inequality inside a cube
I am looking for a reference for the following inequality: if $\Omega \subset [0,1]^d$ satisfies $\mbox{vol}(\Omega) \leq 1/2$, then
$$ \mathcal{H}^{d-1}\left( \partial\Omega \cap (0,1)^d\right) \geq ...
8
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772
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Upper bound of the kissing number in n dimensions
In geometry, a kissing number is defined as the number of
non-overlapping unit spheres that can be arranged such that they each
touch another given unit sphere.
Let $\tau_n$ be the kissing number ...
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3
answers
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Ball ricochetting from a plane of close-packed spheres
Suppose the lower $z \le 0$ halfspace of $\mathbb{R}^3$ is filled with a rigid close-packing of
unit-radius spheres.
(I don't think it matters much for my purposes if it is
an
FCC or an HCP packing.)...
- 151k
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2
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Angle subtended by the shortest segment that bisects the area of a convex polygon
Let $C$ be a convex polygon in the plane and let $s$ be the shortest line segment (I believe this is called a "chord") that divides the area of $C$ in half. What is the smallest angle that $s$ could ...
- 113
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Estimating the Volume of the Metric Polytope
A metric on $n$ points $N$ can be represented as a vector $x \in \mathbb{R}_+^{n \choose 2}$.
For each pair of distinct $i, j \in N$, we have $d(i,j) = d(j,i) = x_{i,j}$. The set of all metrics is ...
- 794
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398
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Möbius-invariant triangle center?
Given any two points x and y on a circle O, one can form four different lenses (regions between two circles, one of which is O) that have corners at x and y and make angles of 2π/3 at their corners. ...
- 18.6k
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Does the surface area of the unit Lp ball go to zero for all $p < \infty$?
We know about volume: The $L_{\infty}$ ball of radius one-half, i.e. the hypercube, has volume $1$ in all dimensions. On the other hand, I believe that for every $1 \leq p < \infty$, the volume of ...
- 4,529
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What must a set of $n$ points in 2D space fulfill so that it is possible to connect them through tangent circles
In high-school I learned how to find the circles that connect points in 2D space forming a curve made out of tangent circles like this:
(The green line shows the "initial direction" of the ...
- 83
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An approach to showing hyperbolic groups are CAT(0)
I've been sitting on this idea for quite a while but I'm not in academia any longer so not likely to ever tackle it on my own. The approach is as follows:
$G$ acts on its boundary $\partial G$
ergo, $...
- 151
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How to interpret couplings in optimal transport?
Let $\mu$ and $\nu$ be two measures on some (at least measurable) space $X$. In optimal transport theory, Monge's problem to
$$ \text{minimize} \quad \int c(x,T(x))\mu(dx) \quad \text{over measurable ...
- 309
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Area method in Lobachevskian geometry
There are many proofs in Euclidean geometry using the area method; for example, Ceva's theorem or the proof of Pythagorean theorem shown below.
Do you know such proofs in hyperbolic geometry?
I ...
- 45k
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Axioms of length
Assume I want to define length of plane curves axiomatically.
It seems to be reasonable to assume that
The length of a unit segment is 1;
Congruent curves have equal lengths;
Length is additive with ...
- 45k
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What is the probability that these sets intersect?
Let $A$ be the subset of $\mathbb{R}^n$ defined by $A=\{x\in\mathbb{R}^{n}:|x_{1}-x_{n}|+\sum_{i=1}^{n-1}|x_{i+1}-x_{i}|\leq d\}$ for a given $d$. Next, sample a point $p$ uniformly in the unit cube, ...
- 4,049
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Bi-Lipschitz version of Kirszbraun's extension theorem
Kirszbraun's theorem for $\mathbb{R}^2$ states the following:
Given any set $S\subset \mathbb{R}^2$ and any Lipschitz function $f:S\rightarrow \mathbb{R}^2$ with Lipschitz constant $k$, $0< k<...
user99087
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Orthonormal bases of R^3 with components lying in the golden field
Greg Egan proved an interesting theorem about unit vectors in $\mathbb{R}^3$ whose components actually lie in the 'golden field' $\mathbb{Q}[\sqrt{5}]$. He found it in our studies of twin dodecahedra:...
- 22.3k
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Embedding Euclidean buildings into products of trees
A Euclidean building has a natural metric space structure. (A definition of Euclidean building can be found on Wikipedia, or, more expansively, in Section 4 of Kleiner-Leeb.)
Question: Is it true ...
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Do elements of the fundamental group give rise to isometries
Let $X$ be a complex algebraic variety, and let $\tilde X\to X$ be its universal cover. Suppose that there exists a Kahler-Einstein metric on $\tilde X$. Note that $\pi_1(X) \subset Aut(\tilde X)$.
...
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Hilbert metric and cross-ratio of points on simplices
Background and motivation:
Consider the cone $C\subset \mathbb{R}^d$ of vectors with non-negative components, and let $\Delta\subset C$ be the simplex of probability vectors (those for which $\sum ...
- 8,903
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Reconstructing an Euclidean point cloud from their pairwise distances
I have a collection of points $P_1, ..., P_N$ in some Euclidean space $\mathbb R^m$ and the coordinates $x_1, x_2, ..., x_N$ respectively associated with them, where $x_i$ is the usual Cartesian tuple ...
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Is the Cheeger constant of an induced subgraph of a cube at most 1?
It is known that the
Cheeger constant
of a
hypercube graph $Q_n$
is exactly $1$, regardless of its dimension $n$. Is $1$ also an upper bound
on the Cheeger constant of nontrivial induced connected ...
- 266
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885
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Maximal tetrahedra inscribed in ellipsoid
Pietro Majer quoted the theorem of Michel Chasles in his MO question,
"Convex curves with many inscribed triangles maximizing perimeter,"
which states that the triangles of maximum perimeter inscribed ...
- 151k
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359
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Can any rectangle be inscribed in any convex figure?
Can any rectangle be inscribed in any convex figure?
- 2,302
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700
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Upper bound for tetrahedron packing?
There have been several recent advances on packing regular tetrahedra in $\mathbb{R}^3$. All the results I've seen have been lower bounds -- first John Conway and Sal Torquato showed that there ...
- 7,857
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Can the algebraic closure of a complete field be complete and of infinite degree?
Yes, this is yet another "foundational" question in valuation theory.
Here's the background: it is a well known classical fact that the dimension (in the purely algebraic sense) of a real ...
- 65.4k
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530
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Whitney's approximation theorem for Lipschitz manifolds
In the smooth setting, Whitney's approximation theorem says the following: If $M,N$ are smooth manifolds and $f,g:M\to N$ are smooth functions that are continuously homotopic (ie there is a continuous ...
- 1,149
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Is there a nonelementary hyperbolic group without this transitivity property?
Some background (see e.g. the books by Ghys & de la Harpe or Bridson & Haefliger for more information):
Let $\Gamma$ be a group with a finite symmetric generating set $S$. Recall that $\Gamma$...
- 2,479
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515
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When a Riemannian manifold with boundary is an Alexandrov space?
Let $M$ be a smooth Riemannian manifold (without boundary). Let $X\subset M$ be a smooth compact submanifold with boundary, $\dim X=\dim M$.
Under what conditions $X$, equipped with the induced ...
- 21.8k
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Translative packing constant strictly larger than lattice packing constant
Simply put, my question is this: what is the smallest dimension, if any,
where we can know for sure that a convex body exists whose translative
packing constant is strictly larger than its lattice ...
- 5,971
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289
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Distortion of tree embedding in Alexandrov spaces
It is a well-known theorem first proved by Bourgain that any map $\varphi:T_n\to H$ from the binary tree of height $n$ to a Hilbert space has distortion at least $C \sqrt{\ln n}$ where $C$ is a ...
- 14.4k
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991
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Higher-order axiomatisations of Euclidean Geometry?
I am currently thinking about the possibility to axiomatise Euclidean Geometry using higher-order axioms. The idea is that all objects are points, and that we only have two primitive notions: A three ...
- 1,228
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490
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Continuous point map for spherical domains
Consider the space $J$ of Jordan domains on the sphere $\textbf{S}^2$, i.e., continuous injective maps from the unit disk into $\textbf{S}^2$ modulo homeomorphisms of the disk. How can one construct a ...
- 7,179
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Symmetries of contractable subsets of $\Bbb R^n$
Let $K\subset\Bbb R^n$ be a non-empty compact subset of $\Bbb R^n$. A symmetry of $K$ is an isometry of $\Bbb R^n$ that fixes $K$ set-wise. Since $K$ is compact, there is always a point $x\in\Bbb R^n$ ...
- 13.6k
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803
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a Littlewood–Offord-type problem concerning the "cubical lattice"
Fix even $n$ and consider the boolean function $f : \{0, 1\}^n \rightarrow \{0, 1\}$, $f : (x_0, \ldots , x_{n - 1}) \mapsto (x_0 \vee x_1) \wedge (x_2 \vee x_3) \wedge \cdots \wedge (x_{n - 2} \vee ...
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Length and curvature for closed curves in negatively curved spaces
In the Euclidean plane, for a closed smooth curve of length $\ell$ whose curvature is bounded above by $\epsilon$ we have the inequality
$$ \ell \ge 2\pi \epsilon^{-1} $$
which follows from the fact ...
- 3,399
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Multidimensional gluing theorem for Riemannian manifolds
I would like to understand whether the following multidimensional (partial) generalization of the A.D. Alexandrov gluing theorem is true and, if yes, whether there is a reference.
(The original ...
- 21.8k
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Differentiability of geodesics in Alexandrov subspaces of Riemannian manifolds
Let $M$ be a smooth Riemannian manifold. Let $X\subset M$ be a closed path connected subset which has curvature bounded below in the sense of Alexandrov with respect to the induced intrinsic metric. ...
- 21.8k
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$C^1$ regularity of harmonic functions on Riemannian manifolds
Consider a smooth, connected and complete Riemannian manifold $M$. It is well known that harmonic functions defined on some open subset of $M$ are $C^\infty$.
I'm interested in knowing whether there ...
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Gromov-Hausdorff convergence for non-compact metric spaces
Let $(X_i,p_i)$, $(X,p)$ be pointed connected proper metric spaces (i.e. the closures of balls are compact). Are the following two statements equivalent?
$\forall r > 0: \bar{B}_r(p_i) \stackrel{...
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Dubins car shortest paths: Decidable?
A Dubins car follows a
Dubins path
in $\mathbb{R}^2$, with constant wheel speed and
limited turning radius.
It is known that the shortest Dubins path in the absence
of obstacles follows circular
arcs ...
- 151k
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416
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Areas of Triangles in (Non-Riemannian) Metric spaces?
I'm looking for a reasonable way to coherently axiomatize both length and area in the absence of a Riemannian structure, i.e., starting only with a metric space; but it's not clear how much of this ...
- 15.5k
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Convex bodies with symmetric shadows
Theorem. If all orthogonal projections of a convex body $K \subset \mathbb{R}^n$ onto $2$-dimensional subspaces have a center of symmetry, then $K$ has a center of symmetry.
This is a classic result ...
- 13.5k
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Polyhedra that combinatorially shadow a sequence
Let $P$ be a polyhedron in $\mathbb{R}^3$.
Say that $P$ combinatorially shadows a sequence of natural numbers $S$ if
there is a continuous rotation of $P$ such that its orthogonal-projection
shadows ...
- 151k
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The rain hull and the rain ridge
Rain falls steadily on an island, a 2-manifold $M$, which you may
assume, as you prefer,
is: (a) smooth, or (b) a PL-manifold, or perhaps even
(c) a
triangulated irregular network (TIN).
After a time,...
- 151k
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minimal diameter of full preimage of torus
Given a set $A\subset \mathbb{R}^n$ such that $A\cap (x+\mathbb{Z}^n)\ne \emptyset$ for any $x\in \mathbb{R}^n$ (that is, $p(A)=\mathbb{T}^n$ for the projection $p:\mathbb{R}^n\rightarrow \mathbb{T}^...
- 109k
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Generalization of Hamiltonian cycles to "Hamiltonian spheres"
One possible generalization of a Hamiltonian cycle in a triangulated plane graph is what could be
called a Hamiltonian sphere: a collection of triangles within a simplicial complex in $\mathbb{R}^3$
...
- 151k
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Coiling Rope in a Box: Decidable?
Is the problem Coiling Rope in a Box decidable? To be specific, is this decidable?
Given $L > 0$ and $r \in (0,\frac{1}{2})$,
both rational,
can a rope of length $L$ and radius $r$
fit ...
- 151k
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1
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556
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A variation on "Hearing the shape of a drum" for polytopes.
Let $\varphi:\mathcal S^{d-1}\longrightarrow \mathbb R_{>0}$ be a strictly positive function describing the boundary $\varphi(\mathbf x)\mathbf x,\mathbf x\in\mathbb S^{d-1}$ of a $d-$dimensional ...
- 17.9k
8
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381
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Estimating flat norm distance from a planar disc
Let $D\subset\mathbb R^2\subset\mathbb R^n$ be a unit planar disc in $\mathbb R^n$. Let $S$ be an orientable two-dimensional surface in $\mathbb R^n$ such that $\partial S=\partial D$. Of course, we ...
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