Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,403 questions
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On area bisectors and perimeter bisectors of planar convex regions
We try to proceed from A claim on the concurrency of area bisectors of planar convex regions
Definitions: Given a planar convex region C, an area bisector of C is any line segment that partitions C ...
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About planar curves on a manifold
I recently came upon the following situation (think of $\mathbb{R}^3$ to simplify): let $S$ be a compact smooth surface with $K>0$ everywhere and define
$$Q=\frac{\sup_{p}\lambda_{1}(p)}{\inf_{p}\...
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'Self-similar and perfect' partitions of planar regions
Definition: A partition of a planar figure into finitely many pieces that are all similar to itself and also mutually non-congruent may be called a self-similar perfect partition.
A classical example ...
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Perfect 'cuboiding' of cubes and cuboids
We try to add a bit to ref 2 listed below. In this post, by 'cuboid', we mean only rectangular cuboids - hexahedra with all faces rectangles and adjacent faces meeting only at right angles. A special ...
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Busemann-Petty type problems on complex vector spaces [closed]
We recently published an article on Busemann-Petty type problems (see https://arxiv.org/abs/2404.05630). As we experienced several times that as soon as an article is published, no updates/corrections/...
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Distance between two convex sets
Setting
If $A$ an $B$ are two symmetric matrices, we denote by $A >B$ when the matrice $A-B$ is definite positive.
In $\left(\mathbb{R}^{*}_{+} \right)^4$, consider the convex set $$ \Lambda = \...
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Sum of upper semi continuous and lower semi continuous functions
Let $X$ be a compact metric space. Assume that $f: X \to \mathbb{R}$ is upper-semi continuous and $g:X \to \mathbb{R}$ is lower semi-continuous. Assume that $\sup \{ f(x)+g(x) : x \in X \}$ is finite. ...
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Simple convergence of convex compact set implies Hausdorff convergence
I am wondering about the following :
In $\mathbb{R}^n$, suppose you are given compact convex bodies $\left\{ C_k : k \geq 1 \right\}$ and $C$, such that for every $x \in \mathbb{R}^n$ $$ \mathbb{1}_{...
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Polyhedra inscribed in a sphere with mutually non-congruent, equal area faces
Two constrained versions of the main question given in this post: Polyhedrons with mutually non-congruent faces, all of equal area. An earlier post that could be related: Cutting a spherical surface ...
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Algorithm to generate configurations with kissing number 12
That the kissing number of a sphere in dimension 3 is 12 is well known. However, it is also known that there is a lot of empty space between the 12 spheres. I deduce (am I wrong?) that there are many ...
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A claim on the largest area circular segment that can be drawn inside a planar convex region
This post adds a little to To find the longest circular arc that can lie inside a given convex polygon
A circular segment is formed by a chord of a circle and the line segment connecting its endpoints....
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$L^p$-compression of metabelian groups
Question: Is there a metabelian group, so that for some $\epsilon >0$ and all $p \in [1, \infty[$ the [equivariant] compression exponent in [any] $L^p$-space is bounded by $1-\epsilon$ (bound does ...
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Can the volume of a neighborhood of the cut locus be arbitrarily small?
Let $(M^n,g)$ be a complete, $n$-dimensional Riemanian manifold without boundary, maybe non-compact. Let $p\in M$ be a point, and $C_p$ the cut locus. It's known that $C_p$ has Hausdorff dimension $\...
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Intersection of unit-norm vectors with a large sum in high dimensions with a spherical cap
Let $d$ and $n$ be integers. For $i \in \lbrace 1,\dots,n \rbrace$ let $x_i \in \mathbb{R}^d$ be a vector such that $\lVert x \rVert=1 $. For a fixed $1/2 < \alpha \leq 1$, assume we have $\lVert \...
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In the limit, do the intersection points of a string figure define a probability measure on the unit disk?
Let D = {z ∈ ℂ | |z| ≤ 1} denote the closed unit disk in the complex plane.
For any integer n ≥ 1 define the nth string figure S(n) ⊂ D as the union of all n(n+1)/2 line segments that connect two ...
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Which polygons allow partition into rational triangles?
A triangle with all side lengths rational is said to be a rational triangle. It is known - for example, Cutting the unit square into pieces with rational length sides - that the unit square allows ...
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Moser iteration epsilon-regularity for non-linear system in general dimension
I am attempting to prove the following result in general dimension $n$. Given $(M^n,g)$ a Riemannian manifold with $\mathrm{Ric}_g \geq -(n-1)$ and $\mathrm{Vol}_g(B_1(x)) \geq v > 0$ for all $x \...
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Intuition behind right-inverse of map from Johnson-Lindenstrauss Lemma
The Johnson–Lindenstrauss lemma states that for every $n$-point subset $X$ of $\mathbb{R}^d$ and each $0<\varepsilon\le 1$, there is a linear map $f:\mathbb{R}^d\to\mathbb{R}^{O(\log(n)/\varepsilon^...
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Planar sections of convex sets in Cartan-Hadamard manifolds
Let $X$ be a convex set in Euclidean space $\mathbf{R}^n$ and $p\in\mathbf{R}^n$ be a fixed point. Then any plane $\Pi$ passing through $p$ intersects $X$ in a convex set. Conversely, this property ...
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Unusual parameterization of the ring Dupin cyclide
I discovered the following by playing with the formulas given in the paper Sculptures in $S^3$ by Schleimer and Segerman.
First, define the following parameterization of the Clifford torus:
$$
p(\...
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Number of polyhedral covers of a triangulation of $S^2$
For a given triangulation (combinatorial Type I. or Type II.) of a $2$-sphere, what is the number of unique polygonal covers with $n$ polygons where ($n$ goes from $2$ to $N$)?
Under polygonal cover, ...
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The optimal embedded and enclosing cardioids for a triangle
Ref: https://en.wikipedia.org/wiki/Cardioid
Earlier posts with similar questions: Smallest 3-ellipses that contain triangles and Curves of constant width that contain triangles
Questions: Given any ...
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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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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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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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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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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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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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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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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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A special configuration of Nine Circles Theorem and Eight Circle Theorem
The result as follows from special configuration of merge Nine Circle Theorem and Eight Circle theorem but it is new:
Problem: Let three circle $(A)$, $(B)$, $(C)$ , let $A_c$ be arbitrary point in ...
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Closed form volumes for intersecting modified cylinders
This question is somewhat related to the question Intersecting cylinders, but where the cylinders are now modified to orbifolds in the hypercube with singularities occurring at the vertices of the ...
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Non-surjective isometries of $l_p$
It is well known that all surjective isometries of $l_p$ for $p\neq 2$ are the signed permutations of the unit vector basis $(e_n)$. Is there a characterization for the linear non-surjective ...
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Plateau problem in the disk: a question about geodesic nets
Consider given a finite collection of points along the boundary of the unit disk $D \subset \mathbf{R}^2$:
\begin{equation}
p_1,\dots,p_{2n} \in \partial D.
\end{equation}
We assume that these are all ...
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Fitting a convex polytope with 𝑛 facets between two nested spheres
This is related to a research problem that is interested in approximation of spheres by convex polytopes.
Let $C_r$ and $C_R$ be two spheres in $\mathbb R^d$ of radius $r$ and $R$, respectively, where ...
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Inside-out dissections of solids
We add to Inside-out dissections of polygons - a generalization. The inside-out (fully inside-out) dissections are defined on pages linked there.
How does one inside-out dissect a tetrahedron into ...
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About the number of faces of the conification of a polytope
Let $P\subset\mathbb{R}^n$ be a polytope of dimension $(n-1)$ such that the origin $\vec{0}\not\in\text{Aff}(P)$, where $\text{Aff}(P)$ denotes the affine hull of $P$ in $\mathbb{R}^n$. Now, we ...
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What is the example of a circle being filled with congruent tiles (not pie slices), with no overlap of the tiles and and no space left?
I think I read somewhere that at one time it was thought the only way to lay tiles that would fill a circle with no overlap of the tiles and no exposed space in the cirlce, was to lay pieces that ...
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Least number of squares of size N that a set of R rectangles can occupy
Given a set $R$ of rectangles of different positive integer sizes, and any number of squares of the same size $N\in\mathbb{N}$, what's the least number of squares $C$ that all the rectangles together ...
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Kissing behavior of planar regions
This post reworks a question that was stated in a slightly different form at Convex region $C$ with least kissing number of copies of $C$.
Background: Given a 2D region $C$ (not necessarily convex), ...
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Uniformly open map on a dense subset
Schauder's lemma asserts that you can always extend a uniformly continuous, uniformly open map from a dense subset of a complete metric space to a uniformly open map on the completion.
I think the ...
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Possible extensions of the perpendicular axes theorem for moment of inertia
This post continues on Moment of inertia from Bisectors and partitioning lines for convex regions defined with respect to the moment of inertia.
The perpendicular axis theorem states that the moment ...
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what belongs in a first university-level geometry course? [closed]
I know this is not really a research question, but I would like to ask it of research mathematicians, to see if there is a consensus. In a recent discussion on this topic, someone suggested that if ...
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About semiconvex and semiconcave functions in geodesically complete CAT(0) spaces
I posted this question a while ago but had no luck. So I'm trying again. Let $(X,d)$ be a proper, geodesically complete $CAT(0)$ space and let $\Sigma_x X$ be the space of directions at a point $x\in ...
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John and Lowner ellipsoid
I am looking at a proof to show that Lowner ellipsoids are unique for centrally symmetric convex body $K$. I want to show basically that
$$
\DeclareMathOperator{\Vol}{Vol}\DeclareMathOperator{\Low}{...
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Lattice packing
Let $\Lambda$ be a lattice in $R^n$ and $R>0$ a real number.
Consider the number $N$ of points in $\Lambda$ of norm less than $R$. Let $R$ goes to infinity. What can be said about the asymptotic ...
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Hyperbolic vs Euclidean balls [closed]
I'm trying to prove that, in the Poincaré half-space of dimension 2, a hyperbolic ball with center $P:=(x,y)$ and radius $r$ is exactly, as a set of points, a euclidean ball with center $(x,y\cosh(r))$...
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All 3-dimensional symmetric reflexive polytopes
$\DeclareMathOperator\Conv{Conv}$I am finding all 3-dimensional symmetric reflexive polytopes. To do so, first, we know that all 2 dim symmetric reflexive polytopes are $X_3=\Conv((-1,-1),(1,0),(0,1))$...
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Multidimensional scaling with partially known distance matrix
As far as I know, multidimensional scaling requires a matrix of pairwise distances between the data points to be available. What if I only have distances between some pairs of points, but not all of ...
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Inside-out dissection
In a recent problem in The College Math Journal (1230) a Heronian triangle is called to have an equivalent rectangle if there exists an integer sided rectangle with the same area and perimeter. For ...
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