Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,249
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On smallest convex m-gons that contain a given n-gon where m<n
Given a convex n-gon region P, and an m less than n, will the least area convex m-gon Q that contains P be such that an edge of Q coincides with an edge of P (in other words Q cannot be such that P ...
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118
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Absolute continuity of the volume growth in a metric space
Let $(M,d)$ be a metric space (separable, complete, better?) and let $\mu$ be a ($\sigma$-additive, positive, locally finite, regular?) Borel measure on $M$. For $x\in M$ consider the volume growth ...
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Name this geometric point?
Is there a formal name for the point which is the reflection of the incenter about the circumcenter of a triangle?
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When do the centers of mass of a uniform convex planar region as a whole and of its boundary alone coincide?
Given a uniform planar convex region C, let us consider 2 centers of mass - the center of mass of the region as a whole and the center of mass of its boundary alone (assuming its boundary to have ...
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Obtaining metric and compatible differential equations on codimension one foliation of $n$-cube
Essentially the same content as this post on Math stack exchange: https://math.stackexchange.com/q/4741835/460999. I don't expect an answer there and after waiting a few days I've decided to post here....
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Asymptotic volume of intersection of n-ball and a cube
Let $B$ be the unit ball in $\mathbb{R}^n$, and let $c\in(0,1)$ be a constant. I'm trying to find the asymptotics for the volume of the intersection $[\frac{c}{\sqrt{n}},1]^n\cap B$ as $n\rightarrow\...
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353
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Geodesic distance on $\mathrm{SO}(n)$
$\DeclareMathOperator\SO{SO}$Recently I came across this old MSE post or this paper (w.o. proof) discussing the geodesic distance on $\SO(n)$ when it is equipped with the left-invariant Riemannian ...
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Finding balls with big measure
Let $(X,d)$ be a compact metric space $n \in \mathbb{N}$ and $\mu$ a finite Borel measure. Suppose there exists $\delta, R>0$ such that for all $0<r<R$.
$$\mu(B(x,r)) < \delta r^n.$$
Under ...
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Understanding $\kappa$-cones
I recently came across the concept of a $\kappa$-cones of a metric space (Chapter I.5.2) of Bridson and Haefliger's book. In their Proposition 5.8, the provide some intuition of $\kappa$-cones by ...
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What do the Carnot groups act on?
My question is in some sense a less ambitious version of the following MO question where the answer was inconclusive.
A Carnot group of step $N$ can be identified within the tensor algebra, modulo ...
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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 ...
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Planar convex region maximizing the difference in 'orientation' between its smallest containing rectangle and largest contained rectangle
We say a rectangle has orientation $\theta$ if the vector from its center to the middle of its shortest side (parallel to the longest side) has some angle $\theta$ with X axis.
Consider a planar ...
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To optimally wrap convex laminae with paper
Ref: On folding a polygonal sheet, Multi-layered wrapping of polyhedra
Basic intent: to wrap a given convex planar lamina with a convex sheet of non-stretchable paper (such that every point on both ...
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What does it mean "parallel"?
I am thinking on a strict definition of the notion of parallel affine sets in a linear space and came to the following
Definition 1: An affine set $A$ is parallel to an affine set $B$ in a linear ...
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A claim on planar sections of 3D convex bodies
Ref: More on shadows of 3D convex bodies,
Shadows and planar sections of polyhedra
Given a 3D convex body C, we define a maximal area (perimeter) section of C with respect to any specified direction $...
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More on shadows of 3D convex bodies
Ref: Shadows and planar sections of polyhedra
By shadow we mean the orthogonal projection of a convex 3D body C onto a 2D plane, for example, the shadow on the xy-plane, with C above (z>0) that ...
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Isometric embedding of 4-element metric spaces into Riemannian manifolds and the curvature
I came across this question Preferred embedding of finite metric spaces in riemaniann manifolds of given dimension. In one of the answers it was stated that it is always possible to isometrically ...
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Relation between the "s" parameter of Ungar's theory of hyperbolic geometry and the eccentricty in the 2D case
In Ungar's theory of hyperbolic geometry for the Minkowski model, there is a parameter $s>0$ which controls the curvature of the hyperbolic segments:
Ungar's theory is not very well-known. An ...
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Is the 3d writhe of ideal knots proportional to their smallest possible 2d writhe?
In a knot, the (two-dimensional) or 2d writhe is the sum of all positive crossings minus the sum of all negative crossings. The 2d writhe is always an integer. There is also, for each knot, a smallest ...
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Integral over quotient of discrete group
Let $Y$ be a proper metric space. By a lattice we mean a discontinuous group of isometries $\Gamma$ with compact quotient $Y/\Gamma$. You may also assume that $\Gamma$ acts freely. Suppose we are ...
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Rolling a sphere on a fractal curve
Given a rectifiable curve C : [0, 1] → R2 in the plane, it makes sense to roll a unit sphere S2 on the plane along that curve and ask what is its net rotation in SO(3).
I wonder if this also makes ...
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A geometric proof that there are infinitely many primes?
Let $e_d$ be the $d$-th standard-basis vector in the Hilbert space $H=l_2(\mathbb{N})$.
Let $h(n) = J_2(n)$ be the second Jordan totient function, defined by:
$$J_2(n) = n^2 \prod_{p|n}(1-1/p^2)$$
...
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Intercept theorem in $\mathbb R^n$
The celebrated intercept theorem(also known as Thales's theorem) provides the ratios between the line segments created when two parallel lines are intercepted by two intersecting lines.
I'm looking ...
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139
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Is fundamental group of a finite volume, negatively curved, cusped manifold a non-uniform lattice?
$\DeclareMathOperator\Mob{Mob}$Some background: (1) A Riemannian manifold $M$ is pinched negatively curved if there is a constant $\tau<\kappa<0$ such that all the sectional curvatures are ...
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Plane curve with continuously increasing Hausdorff dimension
In a recent paper, we required the following fact.
Proposition 1. There exists a simple closed curve $\gamma\subset\mathbb{C}$ with the following property. If $\phi$ is a biholomorphic map, defined on ...
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Is this a smooth approximation to the $\ell$-infinity distance actually a quasi-metric?
The $\|\cdot\|_{\infty}$-norm on $\mathbb{R}^n$ for $n\in \mathbb{Z}^+$ is not a smooth function. However, I came across this post which essentially says that a pointwise approximation to the maximum ...
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Purely non-atomic measure on the Gromov boundary of a finitely generated free group
In the set-up of my previous post, let $\theta$ be a purely non-atomic probability regular measure defined on the Borel $\sigma$-algebra of the metric space $(\partial F, d)$. We say $\theta$ admits a ...
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135
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Name of this geometric point? [closed]
Draw a triangle. At one of the vertices, draw a line through it that bisects the angle. At each of the other two vertices, draw a line through it which is perpendicular to the line that bisects its ...
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Can hyperbolic surfaces approximate every connected compact metric space?
Let $X$ be a connected compact metric space.
Question: Is there a sequence of compact hyperbolic surfaces (the curvature may differ between surfaces) that converges to $X$ in the Gromov-Hausdorff ...
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On families of lines that cut the boundary of a planar convex region in a specified ratio
We proceed from A claim on the concurrency of area bisectors of planar convex regions
This question is somewhat broad.
Background: 'Mathematical Omnibus' by Fuchs and Tabachnikov, Lecture 11 describes ...
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133
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Chains in tilings with the aperiodic monotile
This is starting with any given infinite tiling of the plane by the aperiodic "hat" monotile, where chains of similarly oriented tiles are colored as in Figure 2.2 on p. 10 in the original ...
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Vanishing components of Kähler metric
Let $(X, \omega) $ be a $n$-dimensional complex Kähler manifold such that $\omega^{n-1}=d\alpha $.
Does $\partial\alpha^{n-1,n-2} =0$ (resp. $\bar\partial\alpha^{n-2,n-1} =0$)
Where $\alpha^{n-1,n-2}$ ...
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A claim on the concurrency of area bisectors of planar convex regions
We add a little bit to On 'fair bisectors' of planar convex regions and Bisectors and partitioning lines for convex regions defined with respect to the moment of inertia
Definitions: Given a ...
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Which manhole covers fall through their holes?
Apparently one of the reasons why all manhole covers are shaped like discs is because for any other shape, the manhole cover would fall through its own hole. As stated this is not necessarily a ...
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Generalized Triangle Inequality for Snowflakes
Let $p>0$ and consider a metric space $(X,d)$. I have recently come across a problem where the space $(X,d^q)$ provides is natural; where $q>1$. However, the triangle inquality break (i.e. it ...
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Decidability of completing Penrose tilings
Is the following problem known to be un/decidable? Problem: Given a finite configuration of Penrose tiles in the plane, determine if there is an extension of the configuration tiling the whole plane.
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Quasi-isometry on subsets of manifold
Assume $g$ and $h$ are smooth Riemannian metrics on a manifold $X$ with $d_g$ and $d_h$ the induced distance functions on $X$ (via infimum of length of curves in X connecting two points).
Assume $A$ ...
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Orbit projection geometry
Background:
As shown in [1] and [2], for a closed smooth submanifold $M$ of $\mathbb R^d$, the domain $D_M$ of the projection map $P_M:D_M\rightarrow M$ has a dense interior $\Omega_M$ over which $P_M|...
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Cancellation of elements in the Gromov boundary of a free group
Let $A$ be a finite set of free generators and their inverses and $F$ the free group generated by elements in $A$ (some call $A$ the alphabet of $F$). For each $g\in F$, use $\vert\,g\,\vert$ to ...
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Do cycle graphs embed isometrically in spheres?
I recently came across, what seems to be a folklore. Namely, that cycle graphs embeds isometrically into spheres $S^n(r)$, for some $n\in \mathbb{N}_+$ and some $r>0$. However, I could not track ...
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Kissing number lower bound vs. upper bound - precise meanings?
According to en.wikipedia.org, https://en.wikipedia.org/wiki/Kissing_number#Some_known_bounds
It says the kissing numbers $K$ have lower bound $K_L$ and upper bound $K_S$:
$$
K_L < K < K_U.
$$
I ...
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Leech lattice shortest vector vs other 23 cases and E8 cases
In this paper by Viazovska, she said that:
"The E8-lattice sphere packing 𝒫E8 is the union of open Euclidean balls with centers at
the lattice points and radius $1/\sqrt{2}$." So I think ...
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A regular $n$-gon contains a regular $m$-gon, with $n,m$ coprime, no sides coinciding. What is the maximum number of contact points between them?
A regular $n$-gon contains a regular $m$-gon, where $n$ and $m$ are coprime, with no sides coinciding.
What is the maximum number of contact points between the $n$-gon and the $m$-gon?
(I'm not ...
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323
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Upper bound of covering number of $\ell_1$-ball under $\ell_2$-norm
Let $B^n_1 = \{x : \Vert x\Vert_1 ≤ 1\}$ be the $\ell_1$-norm unit ball. How can we prove the covering of $B^n_1$ under $\Vert\cdot\Vert_2$ satisfies
$$\sqrt{\log N(B^n_1, \Vert\cdot\Vert_2, \epsilon)}...
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Counting fractals modulo "shared complements"
Previously asked at MSE:
Let $\mathscr{H}$ be the space of compact nonempty subsets of $\mathbb{R}^2$ (I'm not especially wedded to dimension $2$, so feel free to tweak that if it would lead to a more ...
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269
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Pythagorean theorem in Riemann metrics of non constant curvature
I already asked the same question here, but received no answer. I was reading this interesting article by Givental
Givental, Alexander. "The Pythagorean theorem: What is it about?" Amer. ...
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All the regular $n$-gons are nested tightly around a unit circle. How to order them to minimize the outer radius, and what is that minimum radius?
Let $u_1,u_2,u_3,\dots$ be a permutation of the integers greater than $2$.
A unit circle is in a regular $u_1$-gon, which is a regular $u_2$-gon, which is in a regular $u_3$-gon, ad infinitum. Each ...
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Mapping a cube to a sphere
I have been looking for a way to map a unit cube (with vertices $x^2=1$, $y^2=1$, $z^2=1$) to a unit sphere ($x^2+y^2+z^2=1$) with minimal distortion of the great circles formed by mapping the ...
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How to find a smallest parallelepiped that bounds the unit ball in a normed space
Consider a finite dimensional normed space $(V,\Vert \cdot \Vert)$. How to find a basis $(e_i)$ of $V$ such that the unit closed ball $\overline B_1$ centered at $0$ is contained in $ P:= \{ x \in V : ...
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633
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A generic metric on $X\cup\mathbb Z$
$\newcommand\abs[1]{\lvert#1\rvert}$Let $(X,d_X)$ be a countable metric space such that $X\cap\mathbb Z=\{0\}$.
Problem. Is there a metric $d$ on the union $Y=X\cup\mathbb Z$ such that
$d(x,y)=d_X(x,...