Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,406 questions
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What does the extension theorem for tilings state?
I have seen several references to the so-called Extension Theorem in the context of tilings of Euclidean space.
E.g. in "The Local Theorem for Monotypic Tilings" one reads
The Extension Theorem [......
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Smallest triangles that contain 2D convex regions with reflection symmetry
Given any 2D convex region $C$ with a mirror symmetry. Two pairs of questions:
We need to find the smallest area (likewise, smallest perimeter) triangle that contains $C$. Is it sufficient to only ...
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Meaning of "quantitative result" [closed]
Recently I've begun reading on metric measure spaces and I keep seeing statements containing the phrase ", quantitatively". What does this mean, I googled it and couldn't find a rigorous answer.
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Known Lipschitz-free spaces
The Lipschitz-Free space (also known as Arens-Eells spaces) $\mathcal{F}(X,d)$ over a pointed metric space $(X,d)$ is a well-studied object. In many instances, we have "concrete" representations of ...
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A source for $01$-polytopes
Can you recommend any books or survey articles on $01$-polytopes, thats is, polytopes with vertices in $\{0,1\}^n$?
I am less interested in random $01$-polytopes, but more in the combinatorial ...
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Inequalities on 4 points in metric spaces with curvature bounded below (CBB)
Let $\kappa > 0$. I have a space $(X,d)$ and take 4 points $w,x,y,z \in X$. I then choose comparison points in the model space $(M_\kappa^2,\bar{d})$ as follows:
Take the comparison triangle $\...
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When are Carnot groups negatively curved and homeomorphic to Euclidean space
When are Carnot groups complete and negatively curved (in the sense of $CAT(\kappa)$ spaces)?
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When are principal lines of curvature geodesics?
Let $S$ be a smooth surface embedded in $\mathbb{R}^3$.
When are (some of) the principal lines of curvature geodesics
on $S$? Perhaps on the ellipsoid below, the (blue) central
cycle, a max principal ...
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How do I find the correct distance spacing for distributing equidistant points on a sphere of a given diameter?
How do I find the correct distance spacing for distributing equidistant points on a sphere of a given diameter?
I don't need to fill the sphere with equidistant points. I just need less than a ...
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A question about a $2^n$-point metric space
For any positive integer $n$, let $X_n$ be the family of all subsets of $\{1,2,\cdots,n\}$.
Let $(X_n,d)$ be the metric space such that
$$d(A,B)=|\,A\triangle B\,|,\ \forall A,B\in X_n$$
where $A\...
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Hausdorff convergence of preimages of discrete-valued functions
Suppose $f_n$, $f:X\to K$ where $K$ is a finite set and $(X,d)$ is a metric space. Suppose also that $f_n(x)\to f(x)$ for all $x\in X$ (pointwise convergence). Finally, let $d_H$ be the Hausdorff ...
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Computing Hausdorff-Gromov distance between these objects
As a bonus question in an exam we were asked to find compact metrix spaces $X,Y$ and $Z$ such that $d_{GH}(X,Y)=d_{GH}(X,Z)=d_{GH}(Y,Z)>0$.
The proposed answer is to take $\{0\},\{-1,1\}$ and $\{-...
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More on divisibility
This is a fuzzier follow-up to this question. Again, we construct the graph whose vertices are integers from $1$ to $n,$ and two vertices are connected whenever one of the corresponding integers ...
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The average length of vectors on spherical caps
Let $U$ denote the uniform distribution on the $n$-ball of radius $1$.
What is the expected square-length of a vector under this distribution:
$$ \mathbf{E}_U[\|x\|^2] $$
By standard concentration-...
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Two geodesics with angle $\pi$ in Alexandrov space
Let two geodesic segments in an Alexandrov space with curvature bounded from below start at the same point and the angle between them equals $\pi$. It is possible that these segments are not the two ...
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Maximum crossings of curvature-constrained curve
Let $C$ be a curve in the plane whose curvature is everywhere $\le 1$.
If $C$ has length $L$, what is the largest number of proper self-crossings
of $C$ as a function of $L$?
For example, the curve ...
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Möbius transformation by 3 points in the Minkowski model
Goal
I'm interested in describing Möbius transformations in the plane, and I'd like to define them in terms of three points and their images.
What I have tried
I know that a projective ...
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Mahler Volume of the Snub 24-Cell
I have been working recently on the Mahler conjecture and I am interested in what the Mahler volume of the snub 24-cell in order to check an example calculation.
The recent paper regarding the snub ...
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Symmetric function
I have a continuous function $\mu(x,y)=G(h_1(x+y), h_2(|x-y|))$ such that $\mu(x,x)=0$ and $\mu(x,y)+\mu(y,z)=\mu(x,z)$ for $x\le y\le z$. I want to show that the only possible case is $\mu(x,y)=c\...
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Higher dimensional convex hull
Let $CH(S)$ be a convex hull of a finite set $S$ and denote the set of all the vertices of $CH(S)$ as $Vert(S)$. For a vertex $v \in Vert(S)$, it has an associated set $E(v)$ which is defined as $E(v)=...
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Planar eucliean bipartite matching with squared distances
This is probably a really stupid question, but suppose I have two sets of points in the plane $X$ and $Y$ each with cardinality $|X| = |Y| = n$. For any bipartite matching $M$ between $X$ and $Y$, ...
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Different Measures On R2
Dear all,
Is there any possible way to construct a set $A \subseteq \mathbb{R}^n $ for which $ H^{n-1} (\partial A) > Leb^ + (A ) $?
Where $ H^{n-1} (\partial A) $ is the Hausdorff measure of the ...
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existence of fractal [closed]
I have a question about fractals;
Suppose $\alpha\in[0,1]$ is real number, is there any fractal $F_\alpha$, such that $Dimension(F_\alpha)=\alpha$?
If yes, do we have any method to construct such ...
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Distances between and among points in a region
Let $X = \{x_1, \dots, x_n\}$ denote a finite set of $n$ points in the unit square $S$, and let's center $S$ at the origin. Let $F(X) = \sum_{i=1}^n \| x_i \| $ and let $G(X) = \iint_S \min_i \|x - ...
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Decomposing a sphere (or defomed sphere) into a vertex-transitive graph with fixed-length curved edges
Please see the original problem specification (which Joseph O'Rourke was responding to in his answer) below.
Motivation -
I'm interested in a particular case of the problem where one wants to ...
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Is there a linear embedding of a simplical 3-complex in R^6?
I've heard that there always is an embedding in $R^7$ (can someone provide a reference for that?) and this number cannot be lowered in general. But I'm interested in a somewhat special case, namely: ...
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Alexandrov's theorem analogue for Galilean kinematics
Let $\mathbb R^4_A$, $\mathbb R^4_B$ be spacetime as seen by two inertial observers $A$, $B$ respectively, and call $f:\mathbb R^4_A \to \mathbb R^4_B$ the change of coordinates.
We assume that $f$ ...
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How to compute the number of regular spheres needed to fill a rectangular space
Computing the volume of a sphere is straightforward 4/3*pi*R^3
As is the volume of a rectangular space length*width*height (e.g. 10*10*6)
How might I go about determining how many spheres would fit ...
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An algorithm to arrange max number of copies of a polygon around and touching another polygon
A related post: To place copies of a planar convex region such that number of 'contacts' among them is maximized
Basic question: Given two convex polygonal regions P and Q, to arrange the max ...
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To place copies of a planar convex region such that number of 'contacts' among them is maximized
A contact between two planar convex regions obviously happens either along a line segment or at a single point.
Question: Given a planar convex region $C$ and a number $n$, we need to lay out $n$ ...
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When "$(\varepsilon,\delta)$-geodesic" cannot be a loop?
EDIT: Let $M$ be a smooth compact Riemannian manifold. Let $\varepsilon,\delta>0$. I call a smooth curve $\gamma\colon [a,b]\to M$ an $(\varepsilon,\delta)$-geodesic if for any $t_1<t_2<t_1+\...
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Divergence functions in hyperbolic groups
Gromov hyperbolicity has many characterizations, one of them being the existence of a super-linear divergence function, see definition below.
We note that in $\mathbb{R}^2$ there is no divergence ...
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How do we calculate the gradient of this function defined using the Riemannian logarithm on a Riemannian manifold?
We consider the following function $\psi$ on an open subset $V\subset M,$ a Riemannian manifold of dimension $m,$ so that $\exp_p:U\to V$ is a diffeomorphism with its inverse $\log_p: V\to U$. Let $v\...
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Qualitative values between two electrons in an atom or how to interpret these values?
This question is a little bit trying to understand physics through geometry of simplex:
Let $E_{i,j}$ be the ionization energy in times the number of hydrogen ionization energy for an element with ...
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Do Gromov hyperbolic spaces admit concical geodesic bicombings?
Consider a metric space $(X,d)$ with a distinguished selection of geodesics, i.e. a geodesic bicombing $\sigma:X\times X\times [0,1]\rightarrow X$. We call a geodesic bicombing conical if it ...
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On largest convex m-gons contained in a given convex n-gon where m < n
This post is the inside-out variant of 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, how to find the max area convex m-gon Q ...
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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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Right-continuity of covering number
Consider an ambient metric space $(\mathcal{X},\Vert\cdot\Vert_\infty)$. Let $\mathcal{B}_1 = \mathcal{B}_{\Vert\cdot\Vert_K}(0,1)\subseteq\mathcal{X}$ be the closed unit ball with respect to some ...
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What properties are preserved by quasi-isometries
Recently, I came across the notion of quasi-isometries, while thinking of "discrete spaces which are surrogates for approximate continuous ones".
What (metric)/geometric properties are ...
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Intersection of conical neighbourhoods on a polyhedral space
Let $P$ be a non-negatively curved (in the Alexandrov sense) polyhedral space (of dimension 3, say), $p,q\in P$ be vertices, and let $e$ be an edge connecting $p$ and $q$. Assume $e$ has cone angle $0&...
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Partitioning convex polygons into quadrilaterals of equal area and perimeter
This post records a little bit more on this question: Partitioning convex polygons into triangles of equal area and perimeter.
The basic question of the above linked post was about this claim: "&...
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Algorithm to find a facet of a polyhedron given the vertices?
I have a set of points $X=\{x_1,\ldots,x_N\}$ on the unit sphere in $\mathbb{R}^d$ ($N>d\ge 3$). What is an algorithm to find any facet of the polyhedron whose vertices are $X$?
The set $X$ has ...
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Approximating the probability of a half-space using random Voronoi diagrams
Fix a half-space $H = \{x_1 \geq 0: ~ (x_1,\dots,x_n) \in \mathbb{R}^n\}$. Let $p$ be a distribution with support in $\mathbb{R}^n$. I am interested in the following way of estimating the weight $p(H) ...
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Generate an ordered set of mostly orthogonal vectors $\{x_i\}$ where $x_i \cdot x_j =0$ iff $\lvert i-j\rvert >m$
I am wondering if there is a way to formulate or generate a matrix $X \in R^{n\times n}$ whose column vectors $\{x_1,\dotsc,x_i,\dotsc,x_n\}$ are such that $x_i$ and $x_j$ are orthogonal iff $\lvert i-...
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Relationships among lattices U14, C2xG23, A15+ and their Delaunay polytopes
Do you have any references explaining the relationships among the lattices U14, C2xG23 aka Q14, and A15+?
Do you have any references explaining the relationships among these lattices and the 7D ...
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Partitioning unit square with equal frequency rectangles
If I had to partition the unit square $[0,1]\times[0,1]$ into $k^2$ rectangles such that the sum of their diagonals is minimum possible, I would simply choose the $k \times k$ grid of squares. Now ...
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Polynomial invariant — from product formula to monomial expansion
Context
This question deals with the polynomial invariant denoted by $ H_{n} $ in Maksym Fedorchuk and Igor Pak's 2004 paper Rigidity and polynomial invariants of convex polytopes (sections 7.6 and 9)....
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A question about Gromov-Lawson construction
We all know that if we consider the connected sum $S^n\# S^n$ of two spheres $S^n$ for $n\geq 3$, then by Gromov-Lawson construction(cf. Gromov, Mikhael; Lawson, H.Blaine Jun., The classification of ...
- 563
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$L^p$-barycenters via continuous selectors
Let $\mathbb{P}$ be a Borel probability measure on $\mathbb{R}^n$ with $\int_{x \in \mathbb{R}^n} \|x\|^p\mathbb{P}(dx)<\infty$. For which $p\in [1,\infty)$ (other than $p=2$) is the following ...
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The relationship between facets of an inscribed polytope and those facets' shadows
I posted this question thinking that the response would be two or three answers that say "Counterexamples to this are found in every textbook—for example this one and this one and this one." ...
